## 16 Polynomials ¶

### 16.1 Introduction to Polynomials ¶

Polynomials are stored in Maxima either in General Form or as Canonical Rational Expressions (CRE) form. The latter is a standard form, and is used internally by operations such as `factor`, `ratsimp`, and so on.

Canonical Rational Expressions constitute a kind of representation which is especially suitable for expanded polynomials and rational functions (as well as for partially factored polynomials and rational functions when `ratfac` is set to `true`). In this CRE form an ordering of variables (from most to least main) is assumed for each expression.

Polynomials are represented recursively by a list consisting of the main variable followed by a series of pairs of expressions, one for each term of the polynomial. The first member of each pair is the exponent of the main variable in that term and the second member is the coefficient of that term which could be a number or a polynomial in another variable again represented in this form. Thus the principal part of the CRE form of `3*x^2-1` is `(X 2 3 0 -1)` and that of `2*x*y+x-3` is `(Y 1 (X 1 2) 0 (X 1 1 0 -3))` assuming `y` is the main variable, and is `(X 1 (Y 1 2 0 1) 0 -3)` assuming `x` is the main variable. "Main"-ness is usually determined by reverse alphabetical order.

The "variables" of a CRE expression needn’t be atomic. In fact any subexpression whose main operator is not `+`, `-`, `*`, `/` or `^` with integer power will be considered a "variable" of the expression (in CRE form) in which it occurs. For example the CRE variables of the expression `x+sin(x+1)+2*sqrt(x)+1` are `x`, `sqrt(x)`, and `sin(x+1)`. If the user does not specify an ordering of variables by using the `ratvars` function Maxima will choose an alphabetic one.

In general, CRE’s represent rational expressions, that is, ratios of polynomials, where the numerator and denominator have no common factors, and the denominator is positive. The internal form is essentially a pair of polynomials (the numerator and denominator) preceded by the variable ordering list. If an expression to be displayed is in CRE form or if it contains any subexpressions in CRE form, the symbol `/R/` will follow the line label.

See the `rat` function for converting an expression to CRE form.

An extended CRE form is used for the representation of Taylor series. The notion of a rational expression is extended so that the exponents of the variables can be positive or negative rational numbers rather than just positive integers and the coefficients can themselves be rational expressions as described above rather than just polynomials. These are represented internally by a recursive polynomial form which is similar to and is a generalization of CRE form, but carries additional information such as the degree of truncation. As with CRE form, the symbol `/T/` follows the line label of such expressions.

Categories: Polynomials · Rational expressions ·

### 16.2 Functions and Variables for Polynomials ¶

Option variable: algebraic ¶

  Default value: `false`

  `algebraic` must be set to `true` in order for the simplification of algebraic integers to take effect.

  Categories: Simplification flags and variables ·

Option variable: berlefact ¶

  Default value: `true`

  When `berlefact` is `false` then the Kronecker factoring algorithm will be used otherwise the Berlekamp algorithm, which is the default, will be used.

  Categories: Polynomials ·

Function: bezout (p1, p2, x) ¶

  an alternative to the `resultant` command. It returns a matrix. `determinant` of this matrix is the desired resultant.

  Examples:

      (%i1) bezout(a*x+b, c*x^2+d, x);
                               [ b c  - a d ]
      (%o1)                    [            ]
                               [  a     b   ]

      (%i2) determinant(%);
                                  2      2
      (%o2)                      a  d + b  c

      (%i3) resultant(a*x+b, c*x^2+d, x);
                                  2      2
      (%o3)                      a  d + b  c

  Categories: Polynomials ·

Function: bothcoef (expr, x) ¶

  Returns a list whose first member is the coefficient of x in expr (as found by `ratcoef` if expr is in CRE form otherwise by `coeff`) and whose second member is the remaining part of expr. That is, `[A, B]` where `expr = A*x + B`.

  Example:

      (%i1) islinear (expr, x) := block ([c],
              c: bothcoef (rat (expr, x), x),
              is (freeof (x, c) and c[1] # 0))$

      (%i2) islinear ((r^2 - (x - r)^2)/x, x);
      (%o2)                         true

  Categories: Polynomials ·

Function: coeff
coeff (expr, x, n)
coeff (expr, x) ¶

  Returns the coefficient of `x^n` in expr, where expr is a polynomial or a monomial term in x. Other than `ratcoef` `coeff` is a strictly syntactical operation and will only find literal instances of `x^n` in the internal representation of expr.

  `coeff(expr, x^n)` is equivalent to `coeff(expr, x, n)`. `coeff(expr, x, 0)` returns the remainder of expr which is free of x. If omitted, n is assumed to be 1.

  x may be a simple variable or a subscripted variable, or a subexpression of expr which comprises an operator and all of its arguments.

  It may be possible to compute coefficients of expressions which are equivalent to expr by applying `expand` or `factor`. `coeff` itself does not apply `expand` or `factor` or any other function.

  `coeff` distributes over lists, matrices, and equations.

  See also ratcoef.

  Examples:

  `coeff` returns the coefficient `x^n` in expr.

      (%i1) coeff (b^3*a^3 + b^2*a^2 + b*a + 1, a^3);
                                      3
      (%o1)                          b

  `coeff(expr, x^n)` is equivalent to `coeff(expr, x, n)`.

      (%i1) coeff (c[4]*z^4 - c[3]*z^3 - c[2]*z^2 + c[1]*z, z, 3);
      (%o1)                         - c
                                       3

      (%i2) coeff (c[4]*z^4 - c[3]*z^3 - c[2]*z^2 + c[1]*z, z^3);
      (%o2)                         - c
                                       3

  `coeff(expr, x, 0)` returns the remainder of expr which is free of x.

      (%i1) coeff (a*u + b^2*u^2 + c^3*u^3, b, 0);
                                  3  3
      (%o1)                      c  u  + a u

  x may be a simple variable or a subscripted variable, or a subexpression of expr which comprises an operator and all of its arguments.

      (%i1) coeff (h^4 - 2*%pi*h^2 + 1, h, 2);
      (%o1)                        - 2 %pi

      (%i2) coeff (v[1]^4 - 2*%pi*v[1]^2 + 1, v[1], 2);
      (%o2)                        - 2 %pi

      (%i3) coeff (sin(1+x)*sin(x) + sin(1+x)^3*sin(x)^3, sin(1+x)^3);
                                      3
      (%o3)                        sin (x)

      (%i4) coeff ((d - a)^2*(b + c)^3 + (a + b)^4*(c - d), a + b, 4);
      (%o4)                         c - d

  `coeff` itself does not apply `expand` or `factor` or any other function.

      (%i1) coeff (c*(a + b)^3, a);
      (%o1)                           0

      (%i2) expand (c*(a + b)^3);
                       3          2        2        3
      (%o2)           b  c + 3 a b  c + 3 a  b c + a  c

      (%i3) coeff (%, a);
                                      2
      (%o3)                        3 b  c

      (%i4) coeff (b^3*c + 3*a*b^2*c + 3*a^2*b*c + a^3*c, (a + b)^3);
      (%o4)                           0

      (%i5) factor (b^3*c + 3*a*b^2*c + 3*a^2*b*c + a^3*c);
                                        3
      (%o5)                      (b + a)  c

      (%i6) coeff (%, (a + b)^3);
      (%o6)                           c

  `coeff` distributes over lists, matrices, and equations.

      (%i1) coeff ([4*a, -3*a, 2*a], a);
      (%o1)                      [4, - 3, 2]

      (%i2) coeff (matrix ([a*x, b*x], [-c*x, -d*x]), x);
                                [  a    b  ]
      (%o2)                     [          ]
                                [ - c  - d ]

      (%i3) coeff (a*u - b*v = 7*u + 3*v, u);
      (%o3)                         a = 7

  Categories: Polynomials ·

Function: content (p_1, x_1, …, x_n) ¶

  Returns a list whose first element is the greatest common divisor of the coefficients of the terms of the polynomial p_1 in the variable x_n (this is the content) and whose second element is the polynomial p_1 divided by the content.

  Examples:

      (%i1) content (2*x*y + 4*x^2*y^2, y);
                                         2
      (%o1)                   [2 x, 2 x y  + y]

  Categories: Polynomials ·

Function: denom (expr) ¶

  Returns the denominator of the rational expression expr.

  See also `num`

      (%i1) g1:(x+2)*(x+1)/((x+3)^2);
                               (x + 1) (x + 2)
      (%o1)                    ---------------
                                         2
                                  (x + 3)

      (%i2) denom(g1);
                                         2
      (%o2)                       (x + 3)

      (%i3) g2:sin(x)/10*cos(x)/y;
                                cos(x) sin(x)
      (%o3)                     -------------
                                    10 y

      (%i4) denom(g2);
      (%o4)                         10 y

  Categories: Expressions ·

Function: divide (p_1, p_2, x_1, …, x_n) ¶

  computes the quotient and remainder of the polynomial p_1 divided by the polynomial p_2, in a main polynomial variable, x_n. The other variables are as in the `ratvars` function. The result is a list whose first element is the quotient and whose second element is the remainder.

  Examples:

      (%i1) divide (x + y, x - y, x);
      (%o1)                       [1, 2 y]

      (%i2) divide (x + y, x - y);
      (%o2)                      [- 1, 2 x]

  Note that `y` is the main variable in the second example.

  Categories: Polynomials ·

Function: eliminate ([eqn_1, …, eqn_n], [x_1, …, x_k]) ¶

  Eliminates variables from equations (or expressions assumed equal to zero) by taking successive resultants. This returns a list of `n - k` expressions with the k variables x_1, …, x_k eliminated. First x_1 is eliminated yielding `n - 1` expressions, then `x_2` is eliminated, etc. If `k = n` then a single expression in a list is returned free of the variables x_1, …, x_k. In this case `solve` is called to solve the last resultant for the last variable.

  Example:

      (%i1) expr1: 2*x^2 + y*x + z;
                                            2
      (%o1)                    z + x y + 2 x

      (%i2) expr2: 3*x + 5*y - z - 1;
      (%o2)                  - z + 5 y + 3 x - 1

      (%i3) expr3: z^2 + x - y^2 + 5;
                                2    2
      (%o3)                    z  - y  + x + 5

      (%i4) eliminate ([expr3, expr2, expr1], [y, z]);
                   8         7         6          5          4
      (%o4) [7425 x  - 1170 x  + 1299 x  + 12076 x  + 22887 x
                                          3         2
                                  - 5154 x  - 1291 x  + 7688 x + 15376]

  Categories: Polynomials · Algebraic equations ·

Function: ezgcd (p_1, p_2, p_3, …) ¶

  Returns a list whose first element is the greatest common divisor of the polynomials p_1, p_2, p_3, … and whose remaining elements are the polynomials divided by the greatest common divisor. This always uses the `ezgcd` algorithm.

  See also `gcd`, `gcdex`, `gcdivide`, and poly_gcd.

  Examples:

  The three polynomials have the greatest common divisor `2*x-3`. The gcd is first calculated with the function `gcd` and then with the function `ezgcd`.

      (%i1) p1 : 6*x^3-17*x^2+14*x-3;
                              3       2
      (%o1)                6 x  - 17 x  + 14 x - 3

      (%i2) p2 : 4*x^4-14*x^3+12*x^2+2*x-3;
                          4       3       2
      (%o2)            4 x  - 14 x  + 12 x  + 2 x - 3

      (%i3) p3 : -8*x^3+14*x^2-x-3;
                                3       2
      (%o3)                - 8 x  + 14 x  - x - 3

      (%i4) gcd(p1, gcd(p2, p3));
      (%o4)                        2 x - 3

      (%i5) ezgcd(p1, p2, p3);
                         2               3      2           2
      (%o5) [2 x - 3, 3 x  - 4 x + 1, 2 x  - 4 x  + 1, - 4 x  + x + 1]

  Categories: Polynomials ·

Option variable: facexpand ¶

  Default value: `true`

  `facexpand` controls whether the irreducible factors returned by `factor` are in expanded (the default) or recursive (normal CRE) form.

  Categories: Polynomials ·

Function: factor
factor (expr)
factor (expr, p) ¶

  Factors the expression expr, containing any number of variables or functions, into factors irreducible over the integers. `factor (expr, p)` factors expr over the field of rationals with an element adjoined whose minimum polynomial is p.

  `factor` uses `ifactors` function for factoring integers.

  `factorflag` if `false` suppresses the factoring of integer factors of rational expressions.

  `dontfactor` may be set to a list of variables with respect to which factoring is not to occur. (It is initially empty). Factoring also will not take place with respect to any variables which are less important (using the variable ordering assumed for CRE form) than those on the `dontfactor` list.

  `savefactors` if `true` causes the factors of an expression which is a product of factors to be saved by certain functions in order to speed up later factorizations of expressions containing some of the same factors.

  `berlefact` if `false` then the Kronecker factoring algorithm will be used otherwise the Berlekamp algorithm, which is the default, will be used.

  `intfaclim` if `true` maxima will give up factorization of integers if no factor is found after trial divisions and Pollard’s rho method. If set to `false` (this is the case when the user calls `factor` explicitly), complete factorization of the integer will be attempted. The user’s setting of `intfaclim` is used for internal calls to `factor`. Thus, `intfaclim` may be reset to prevent Maxima from taking an inordinately long time factoring large integers.

  `factor_max_degree` if set to a positive integer `n` will prevent certain polynomials from being factored if their degree in any variable exceeds `n`.

  See also `collectterms` and `sqfr`

  Examples:

      (%i1) factor (2^63 - 1);
                          2
      (%o1)              7  73 127 337 92737 649657

      (%i2) factor (-8*y - 4*x + z^2*(2*y + x));
      (%o2)               (2 y + x) (z - 2) (z + 2)

      (%i3) -1 - 2*x - x^2 + y^2 + 2*x*y^2 + x^2*y^2;
                      2  2        2    2    2
      (%o3)          x  y  + 2 x y  + y  - x  - 2 x - 1

      (%i4) block ([dontfactor: [x]], factor (%/36/(1 + 2*y + y^2)));
                             2
                           (x  + 2 x + 1) (y - 1)
      (%o4)                ----------------------
                                 36 (y + 1)

      (%i5) factor (1 + %e^(3*x));
                            x         2 x     x
      (%o5)              (%e  + 1) (%e    - %e  + 1)

      (%i6) factor (1 + x^4, a^2 - 2);
                          2              2
      (%o6)             (x  - a x + 1) (x  + a x + 1)

      (%i7) factor (-y^2*z^2 - x*z^2 + x^2*y^2 + x^3);
                             2
      (%o7)              - (y  + x) (z - x) (z + x)

      (%i8) (2 + x)/(3 + x)/(b + x)/(c + x)^2;
                                   x + 2
      (%o8)               ------------------------
                                                 2
                          (x + 3) (x + b) (x + c)

      (%i9) ratsimp (%);
                      4                  3
      (%o9) (x + 2)/(x  + (2 c + b + 3) x
           2                       2             2                   2
       + (c  + (2 b + 6) c + 3 b) x  + ((b + 3) c  + 6 b c) x + 3 b c )

      (%i10) partfrac (%, x);
                 2                   4                3
      (%o10) - (c  - 4 c - b + 6)/((c  + (- 2 b - 6) c
           2              2         2                2
       + (b  + 12 b + 9) c  + (- 6 b  - 18 b) c + 9 b ) (x + c))
                       c - 2
       - ---------------------------------
           2                             2
         (c  + (- b - 3) c + 3 b) (x + c)
                               b - 2
       + -------------------------------------------------
                   2             2       3      2
         ((b - 3) c  + (6 b - 2 b ) c + b  - 3 b ) (x + b)
                               1
       - ----------------------------------------------
                   2
         ((b - 3) c  + (18 - 6 b) c + 9 b - 27) (x + 3)

      (%i11) map ('factor, %);
                    2
                   c  - 4 c - b + 6                 c - 2
      (%o11) - ------------------------- - ------------------------
                      2        2                                  2
               (c - 3)  (c - b)  (x + c)   (c - 3) (c - b) (x + c)
                             b - 2                        1
                  + ------------------------ - ------------------------
                                   2                          2
                    (b - 3) (c - b)  (x + b)   (b - 3) (c - 3)  (x + 3)

      (%i12) ratsimp ((x^5 - 1)/(x - 1));
                             4    3    2
      (%o12)                x  + x  + x  + x + 1

      (%i13) subst (a, x, %);
                             4    3    2
      (%o13)                a  + a  + a  + a + 1

      (%i14) factor (%th(2), %);
                             2        3        3    2
      (%o14)   (x - a) (x - a ) (x - a ) (x + a  + a  + a + 1)

      (%i15) factor (1 + x^12);
                             4        8    4
      (%o15)               (x  + 1) (x  - x  + 1)

      (%i16) factor (1 + x^99);
                       2            6    3
      (%o16) (x + 1) (x  - x + 1) (x  - x  + 1)
         10    9    8    7    6    5    4    3    2
       (x   - x  + x  - x  + x  - x  + x  - x  + x  - x + 1)
         20    19    17    16    14    13    11    10    9    7    6
       (x   + x   - x   - x   + x   + x   - x   - x   - x  + x  + x
          4    3            60    57    51    48    42    39    33
       - x  - x  + x + 1) (x   + x   - x   - x   + x   + x   - x
          30    27    21    18    12    9    3
       - x   - x   + x   + x   - x   - x  + x  + 1)

  Categories: Polynomials ·

Option variable: factor_max_degree ¶

  Default value: `1000`

  When `factor_max_degree` is set to a positive integer `n`, it will prevent Maxima from attempting to factor certain polynomials whose degree in any variable exceeds `n`. If `factor_max_degree_print_warning` is true, a warning message will be printed. `factor_max_degree` can be used to prevent excessive memory usage and/or computation time and stack overflows. Note that "obvious" factoring of polynomials such as `x^2000+x^2001` to `x^2000*(x+1)` will still take place. To disable this behavior, set `factor_max_degree` to `0`.

  Example:

      (%i1) factor_max_degree : 100$

      (%i2) factor(x^100-1);
                              2        4    3    2
      (%o2) (x - 1) (x + 1) (x  + 1) (x  - x  + x  - x + 1)
         4    3    2            8    6    4    2
       (x  + x  + x  + x + 1) (x  - x  + x  - x  + 1)
         20    15    10    5        20    15    10    5
       (x   - x   + x   - x  + 1) (x   + x   + x   + x  + 1)
         40    30    20    10
       (x   - x   + x   - x   + 1)

      (%i3) factor(x^101-1);
                                     101
      Refusing to factor polynomial x    - 1
                     because its degree exceeds factor_max_degree (100)
                                   101
      (%o3)                       x    - 1

  See also: `factor_max_degree_print_warning`

  Categories: Polynomials ·

Option variable: factor_max_degree_print_warning ¶

  Default value: `true`

  When factor_max_degree_print_warning is true, then Maxima will print a warning message when the factoring of a polynomial is prevented because its degree exceeds the value of factor_max_degree.

  See also: `factor_max_degree`

  Categories: Polynomials ·

Option variable: factorflag ¶

  Default value: `false`

  When `factorflag` is `false`, suppresses the factoring of integer factors of rational expressions.

  Categories: Polynomials ·

Function: factorout (expr, x_1, x_2, …) ¶

  Rearranges the sum expr into a sum of terms of the form `f (x_1, x_2, …)*g` where `g` is a product of expressions not containing any x_i and `f` is factored.

  Note that the option variable `keepfloat` is ignored by `factorout`.

  Example:

      (%i1) expand (a*(x+1)*(x-1)*(u+1)^2);
                   2  2          2      2      2
      (%o1)     a u  x  + 2 a u x  + a x  - a u  - 2 a u - a

      (%i2) factorout(%,x);
               2
      (%o2) a u  (x - 1) (x + 1) + 2 a u (x - 1) (x + 1)
                                                    + a (x - 1) (x + 1)

  Categories: Expressions ·

Function: factorsum (expr) ¶

  Tries to group terms in factors of expr which are sums into groups of terms such that their sum is factorable. `factorsum` can recover the result of `expand ((x + y)^2 + (z + w)^2)` but it can’t recover `expand ((x + 1)^2 + (x + y)^2)` because the terms have variables in common.

  Example:

      (%i1) expand ((x + 1)*((u + v)^2 + a*(w + z)^2));
                 2      2                            2      2
      (%o1) a x z  + a z  + 2 a w x z + 2 a w z + a w  x + v  x
                                           2        2    2            2
                              + 2 u v x + u  x + a w  + v  + 2 u v + u

      (%i2) factorsum (%);
                                         2          2
      (%o2)            (x + 1) (a (z + w)  + (v + u) )

  Categories: Expressions ·

Function: fasttimes (p_1, p_2) ¶

  Returns the product of the polynomials p_1 and p_2 by using a special algorithm for multiplication of polynomials. `p_1` and `p_2` should be multivariate, dense, and nearly the same size. Classical multiplication is of order `n_1 n_2` where `n_1` is the degree of `p_1` and `n_2` is the degree of `p_2`. `fasttimes` is of order `max (n_1, n_2)^1.585`.

  Categories: Polynomials ·

Function: fullratsimp (expr) ¶

  `fullratsimp` repeatedly applies `ratsimp` followed by non-rational simplification to an expression until no further change occurs, and returns the result.

  When non-rational expressions are involved, one call to `ratsimp` followed as is usual by non-rational ("general") simplification may not be sufficient to return a simplified result. Sometimes, more than one such call may be necessary. `fullratsimp` makes this process convenient.

  `fullratsimp (expr, x_1, ..., x_n)` takes one or more arguments similar to `ratsimp` and `rat`.

  Example:

      (%i1) expr: (x^(a/2) + 1)^2*(x^(a/2) - 1)^2/(x^a - 1);
                             a/2     2   a/2     2
                           (x    - 1)  (x    + 1)
      (%o1)                -----------------------
                                    a
                                   x  - 1

      (%i2) ratsimp (expr);
                                2 a      a
                               x    - 2 x  + 1
      (%o2)                    ---------------
                                    a
                                   x  - 1

      (%i3) fullratsimp (expr);
                                    a
      (%o3)                        x  - 1

      (%i4) rat (expr);
                             a/2 4       a/2 2
                           (x   )  - 2 (x   )  + 1
      (%o4)/R/             -----------------------
                                    a
                                   x  - 1

  Categories: Simplification functions · Rational expressions ·

Function: fullratsubst (new, old, expr)
fullratsubst (`old = new`, expr)
fullratsubst (`[ old_1 = new_1, …, old_n = new_n ]`, expr) ¶

  `fullratsubst` applies `lratsubst` repeatedly until expr stops changing (or `lrats_max_iter` is reached). This function is useful when the replacement expression and the replaced expression have one or more variables in common.

  `fullratsubst` accepts its arguments in the format of `ratsubst` or `lratsubst`.

  Examples:

  - `subst` can carry out multiple substitutions. `lratsubst` is analogous to `subst`.

      (%i1) subst ([a = b, c = d], a + c);
      (%o1)                         d + b

      (%i2) lratsubst ([a^2 = b, c^2 = d], (a + e)*c*(a + c));
      (%o2)                (d + a c) e + a d + b c

  - If only one substitution is desired, then a single equation may be given as first argument.

      (%i1) lratsubst (a^2 = b, a^3);
      (%o1)                          a b

  - `fullratsubst` is equivalent to `ratsubst` except that it recurses until its result stops changing.

      (%i1) ratsubst (b*a, a^2, a^3);
                                     2
      (%o1)                         a  b

      (%i2) fullratsubst (b*a, a^2, a^3);
                                       2
      (%o2)                         a b

  - `fullratsubst` also accepts a list of equations or a single equation as first argument.

      (%i1) fullratsubst ([a^2 = b, b^2 = c, c^2 = a], a^3*b*c);
      (%o1)                           b

      (%i2) fullratsubst (a^2 = b*a, a^3);
                                       2
      (%o2)                         a b

  - `fullratsubst` catches potential infinite recursions. lrats_max_iter.

  Warning: fullratsubst2(listofeqns,expr): reached maximum iterations of 15 . Increase ‘lrats_max_iter’ to increase this limit.

      (%i1) fullratsubst (b*a^2, a^2, a^3), lrats_max_iter=15;
                                    3  15
      (%o1)                        a  b

  See also `lrats_max_iter` and `fullratsubstflag`.

  Categories: Rational expressions ·

Option variable: fullratsubstflag ¶

  Default value: `false`

  An option variable that is set to `true` in `fullratsubst`.

  Categories: Polynomials · Rational expressions ·

Function: gcd (p_1, p_2, x_1, …) ¶

  Returns the greatest common divisor of p_1 and p_2. The flag `gcd` determines which algorithm is employed. Setting `gcd` to `ez`, `subres`, `red`, or `spmod` selects the `ezgcd`, subresultant `prs`, reduced, or modular algorithm, respectively. If `gcd` `false` then `gcd (p_1, p_2, x)` always returns 1 for all x. Many functions (e.g. `ratsimp`, `factor`, etc.) cause gcd’s to be taken implicitly. For homogeneous polynomials it is recommended that `gcd` equal to `subres` be used. To take the gcd when an algebraic is present, e.g., `gcd (x^2 - 2*sqrt(2)* x + 2, x - sqrt(2))`, the option variable `algebraic` must be `true` and `gcd` must not be `ez`.

  The `gcd` flag, default: `spmod`, if `false` will also prevent the greatest common divisor from being taken when expressions are converted to canonical rational expression (CRE) form. This will sometimes speed the calculation if gcds are not required.

  See also `ezgcd`, `gcdex`, `gcdivide`, and poly_gcd.

  Example:

      (%i1) p1:6*x^3+19*x^2+19*x+6;
                              3       2
      (%o1)                6 x  + 19 x  + 19 x + 6

      (%i2) p2:6*x^5+13*x^4+12*x^3+13*x^2+6*x;
                        5       4       3       2
      (%o2)          6 x  + 13 x  + 12 x  + 13 x  + 6 x

      (%i3) gcd(p1, p2);
                                  2
      (%o3)                    6 x  + 13 x + 6

      (%i4) p1/gcd(p1, p2), ratsimp;
      (%o4)                         x + 1

      (%i5) p2/gcd(p1, p2), ratsimp;
                                    3
      (%o5)                        x  + x

  `ezgcd` returns a list whose first element is the greatest common divisor of the polynomials p_1 and p_2, and whose remaining elements are the polynomials divided by the greatest common divisor.

      (%i1) p1:6*x^3+19*x^2+19*x+6 $
      (%i2) p2:6*x^5+13*x^4+12*x^3+13*x^2+6*x $

      (%i3) ezgcd(p1, p2);
                          2                     3
      (%o3)           [6 x  + 13 x + 6, x + 1, x  + x]

  Categories: Polynomials · Rational expressions ·

Function: gcdex
gcdex (f, g)
gcdex (f, g, x) ¶

  Returns a list `[a, b, u]` where u is the greatest common divisor (gcd) of f and g, and u is equal to `a f + b g`. The arguments f and g should be univariate polynomials, or else polynomials in x a supplied main variable since we need to be in a principal ideal domain for this to work. The gcd means the gcd regarding f and g as univariate polynomials with coefficients being rational functions in the other variables.

  `gcdex` implements the Euclidean algorithm, where we have a sequence of `L[i]: [a[i], b[i], r[i]]` which are all perpendicular to `[f, g, -1]` and the next one is built as if `q = quotient(r[i]/r[i+1])` then `L[i+2]: L[i] - q L[i+1]`, and it terminates at `L[i+1]` when the remainder `r[i+2]` is zero.

  The arguments f and g can be integers. For this case the function `igcdex` is called by `gcdex`.

  See also `ezgcd`, `gcd`, `gcdivide`, and poly_gcd.

  Examples:

      (%i1) gcdex (x^2 + 1, x^3 + 4);
                             2
                            x  + 4 x - 1  x + 4
      (%o1)/R/           [- ------------, -----, 1]
                                 17        17

      (%i2) % . [x^2 + 1, x^3 + 4, -1];
      (%o2)/R/                        0

  Note that the gcd in the following is `1` since we work in `k(y)[x]`, not the `y+1` we would expect in `k[y, x]`.

      (%i1) gcdex (x*(y + 1), y^2 - 1, x);
                                     1
      (%o1)/R/                 [0, ------, 1]
                                    2
                                   y  - 1

  Categories: Polynomials · Rational expressions ·

Function: gcfactor (n) ¶

  Factors the Gaussian integer n over the Gaussian integers, i.e., numbers of the form `a + b `%i`` where a and b are rational integers (i.e., ordinary integers). Factors are normalized by making a and b non-negative.

  Categories: Integers ·

Function: gfactor (expr) ¶

  Factors the polynomial expr over the Gaussian integers (that is, the integers with the imaginary unit `%i` adjoined). This is like `factor (expr, a^2+1)` where a is `%i`.

  Example:

      (%i1) gfactor (x^4 - 1);
      (%o1)           (x - 1) (x + 1) (x - %i) (x + %i)

  Categories: Polynomials ·

Function: gfactorsum (expr) ¶

  is similar to `factorsum` but applies `gfactor` instead of `factor`.

  Categories: Expressions ·

Function: hipow (expr, x) ¶

  Returns the highest explicit exponent of x in expr. x may be a variable or a general expression. If x does not appear in expr, `hipow` returns `0`.

  `hipow` does not consider expressions equivalent to `expr`. In particular, `hipow` does not expand `expr`, so `hipow (expr, x)` and `hipow (expand (expr, x))` may yield different results.

  Examples:

      (%i1) hipow (y^3 * x^2 + x * y^4, x);
      (%o1)                           2

      (%i2) hipow ((x + y)^5, x);
      (%o2)                           1

      (%i3) hipow (expand ((x + y)^5), x);
      (%o3)                           5

      (%i4) hipow ((x + y)^5, x + y);
      (%o4)                           5

      (%i5) hipow (expand ((x + y)^5), x + y);
      (%o5)                           0

  Categories: Expressions ·

Option variable: intfaclim ¶

  Default value: true

  If `true`, maxima will give up factorization of integers if no factor is found after trial divisions and Pollard’s rho method and factorization will not be complete.

  When `intfaclim` is `false` (this is the case when the user calls `factor` explicitly), complete factorization will be attempted. `intfaclim` is set to `false` when factors are computed in `divisors`, `divsum` and `totient`.

  Internal calls to `factor` respect the user-specified value of `intfaclim`. Setting `intfaclim` to `true` may reduce the time spent factoring large integers.

  Categories: Integers ·

Option variable: keepfloat ¶

  Default value: `false`

  When `keepfloat` is `true`, prevents floating point numbers from being rationalized when expressions which contain them are converted to canonical rational expression (CRE) form.

  Note that the function `solve` and those functions calling it (`eigenvalues`, for example) currently ignore this flag, converting floating point numbers anyway.

  Examples:

      (%i1) rat(x/2.0);
      rat: replaced 0.5 by 1/2 = 0.5
                                      x
      (%o1)/R/                        -
                                      2

      (%i2) rat(x/2.0), keepfloat;
      (%o2)/R/                      0.5 x

  `solve` ignores `keepfloat`:

      (%i1) solve(1.0-x,x), keepfloat;
      rat: replaced 1.0 by 1/1 = 1.0
      (%o1)                        [x = 1]

  Categories: Numerical evaluation ·

Function: lopow (expr, x) ¶

  Returns the lowest exponent of x which explicitly appears in expr. Thus

      (%i1) lopow ((x+y)^2 + (x+y)^a, x+y);
      (%o1)                       min(2, a)

  Categories: Expressions ·

Function: lratsubst
lratsubst (new, old, expr)
lratsubst (`old = new`, expr)
lratsubst (`[ old_1 = new_1, …, old_n = new_n ]`, expr) ¶

  `lratsubst` is analogous to `subst` except that it uses `ratsubst` to perform substitutions.

  When `lratsubst` is given two arguments, the first argument may be an equation, a list of equations, or a list of exactly one element, which is a list of equations.

  Substitutions are made in the order given by the list of equations, that is, from left to right.

  Multiple substitutions are serial, not parallel. That is, later substitutions are performed on the results of earlier ones.

  When `lratsubst` is given three arguments new, old, and expr, it is equivalent to `lratsubst(old = new, expr)`.

  Examples:

  `lratsubst` can carry out multiple substitutions. `lratsubst` is analogous to `subst`.

      (%i1) lratsubst ([a = b, c = d], a + c);
      (%o1)                         d + b

      (%i2) lratsubst ([a^2 = b, c^2 = d], (a + e)*c*(a + c));
      (%o2)                (d + a c) e + a d + b c

  If only one substitution is desired, then a single equation may be given as first argument.

      (%i1) lratsubst (a^2 = b, a^3);
      (%o1)                          a b

  The first argument may be a list of exactly one element, which is a list of equations.

      (%i1) lratsubst ([[a^2=b*a, b=c]], a^3);
                                     2
      (%o1)                         a  c

  See also `fullratsubst`.

  Categories: Polynomials · Rational expressions ·

Option variable: lrats_max_iter ¶

  Default value: `100000`

  The upper limit on the number of iterations that `fullratsubst` and `lratsubst` may perform. It must be set to a positive integer. See the example for `fullratsubst`.

  Categories: Polynomials · Rational expressions ·

Option variable: modulus ¶

  Default value: `false`

  When `modulus` is a positive number p, operations on canonical rational expressions (CREs, as returned by `rat` and related functions) are carried out modulo p, using the so-called "balanced" modulus system in which `n modulo p` is defined as an integer k in `[-(p-1)/2, ..., 0, ..., (p-1)/2]` when p is odd, or `[-(p/2 - 1), ..., 0, ...., p/2]` when p is even, such that `a p + k` equals n for some integer a.

  If expr is already in canonical rational expression (CRE) form when `modulus` is reset, then you may need to re-rat expr, e.g., `expr: rat (ratdisrep (expr))`, in order to get correct results.

  Typically `modulus` is set to a prime number. If `modulus` is set to a positive non-prime integer, this setting is accepted, but a warning message is displayed. Maxima signals an error, when zero or a negative integer is assigned to `modulus`.

  Examples:

      (%i1) modulus:7;
      (%o1)                           7

      (%i2) polymod([0,1,2,3,4,5,6,7]);
      (%o2)            [0, 1, 2, 3, - 3, - 2, - 1, 0]

      (%i3) modulus:false;
      (%o3)                         false

      (%i4) poly:x^6+x^2+1;
                                  6    2
      (%o4)                      x  + x  + 1

      (%i5) factor(poly);
                                  6    2
      (%o5)                      x  + x  + 1

      (%i6) modulus:13;
      (%o6)                          13

      (%i7) factor(poly);
                            2        4      2
      (%o7)               (x  + 6) (x  - 6 x  - 2)

      (%i8) polymod(%);
                                  6    2
      (%o8)                      x  + x  + 1

  Categories: Integers ·

Function: num (expr) ¶

  Returns the numerator of expr if it is a ratio. If expr is not a ratio, expr is returned.

  `num` evaluates its argument.

  See also `denom`

      (%i1) g1:(x+2)*(x+1)/((x+3)^2);
                               (x + 1) (x + 2)
      (%o1)                    ---------------
                                         2
                                  (x + 3)

      (%i2) num(g1);
      (%o2)                    (x + 1) (x + 2)

      (%i3) g2:sin(x)/10*cos(x)/y;
                                cos(x) sin(x)
      (%o3)                     -------------
                                    10 y

      (%i4) num(g2);
      (%o4)                     cos(x) sin(x)

  Categories: Expressions ·

Function: polydecomp (p, x) ¶

  Decomposes the polynomial p in the variable x into the functional composition of polynomials in x. `polydecomp` returns a list `[p_1, ..., p_n]` such that

      lambda ([x], p_1) (lambda ([x], p_2) (... (lambda ([x], p_n) (x))
        ...))

  is equal to p. The degree of p_i is greater than 1 for i less than n.

  Such a decomposition is not unique.

  Examples:

      (%i1) polydecomp (x^210, x);
                                7   5   3   2
      (%o1)                   [x , x , x , x ]

      (%i2) p : expand (subst (x^3 - x - 1, x, x^2 - a));
                      6      4      3    2
      (%o2)          x  - 2 x  - 2 x  + x  + 2 x - a + 1

      (%i3) polydecomp (p, x);
                              2       3
      (%o3)                 [x  - a, x  - x - 1]

  The following function composes `L = [e_1, ..., e_n]` as functions in `x`; it is the inverse of polydecomp:

      (%i1) compose (L, x) :=
        block ([r : x], for e in L do r : subst (e, x, r), r) $

  Re-express above example using `compose`:

      (%i1) compose (L, x) :=
        block ([r : x], for e in L do r : subst (e, x, r), r) $

      (%i2) polydecomp (compose ([x^2 - a, x^3 - x - 1], x), x);
                              2       3
      (%o2)                 [x  - a, x  - x - 1]

  Note that though `compose (polydecomp (p, x), x)` always returns p (unexpanded), `polydecomp (compose ([p_1, ..., p_n], x), x)` does not necessarily return `[p_1, ..., p_n]`:

      (%i1) compose (L, x) :=
        block ([r : x], for e in L do r : subst (e, x, r), r) $

      (%i2) polydecomp (compose ([x^2 + 2*x + 3, x^2], x), x);
                                2       2
      (%o2)                   [x  + 2, x  + 1]

      (%i3) polydecomp (compose ([x^2 + x + 1, x^2 + x + 1], x), x);
                            2       2
                           x  + 3  x  + 5
      (%o3)               [------, ------, 2 x + 1]
                             4       2

  Categories: Polynomials ·

Function: polymod
polymod (p)
polymod (p, m) ¶

  Converts the polynomial p to a modular representation with respect to the current modulus which is the value of the variable `modulus`.

  `polymod (p, m)` specifies a modulus m to be used instead of the current value of `modulus`.

  See modulus.

  Categories: Polynomials ·

Function: polynomialp
polynomialp (p, L, coeffp, exponp)
polynomialp (p, L, coeffp)
polynomialp (p, L) ¶

  Return `true` if p is a polynomial in the variables in the list L. The predicate coeffp must evaluate to `true` for each coefficient, and the predicate exponp must evaluate to `true` for all exponents of the variables in L. If you want to use a non-default value for exponp, you must supply coeffp with a value even if you want to use the default for coeffp.

  The command `polynomialp (p, L, coeffp)` is equivalent to `polynomialp (p, L, coeffp, 'nonnegintegerp)` and the command `polynomialp (p, L)` is equivalent to `polynomialp (p, L, 'constantp, 'nonnegintegerp)`.

  The polynomial needn’t be expanded:

      (%i1) polynomialp ((x + 1)*(x + 2), [x]);
      (%o1)                         true

      (%i2) polynomialp ((x + 1)*(x + 2)^a, [x]);
      (%o2)                         false

  An example using non-default values for coeffp and exponp:

      (%i1) polynomialp ((x + 1)*(x + 2)^(3/2), [x], numberp, numberp);
      (%o1)                         true

      (%i2) polynomialp ((x^(1/2) + 1)*(x + 2)^(3/2), [x], numberp,
                                                             numberp);
      (%o2)                         true

  Polynomials with two variables:

      (%i1) polynomialp (x^2 + 5*x*y + y^2, [x]);
      (%o1)                         false

      (%i2) polynomialp (x^2 + 5*x*y + y^2, [x, y]);
      (%o2)                         true

  Polynomial in one variable and accepting any expression free of `x` as a coefficient.

      (%i1) polynomialp (a*x^2 + b*x + c, [x]);
      (%o1)                         false

      (%i2) polynomialp (a*x^2 + b*x + c, [x], lambda([ex], freeof(x, ex)));
      (%o2)                         true

  Categories: Predicate functions · Polynomials ·

Function: quotient
quotient (p_1, p_2)
quotient (p_1, p_2, x_1, …, x_n) ¶

  Returns the polynomial p_1 divided by the polynomial p_2. The arguments x_1, …, x_n are interpreted as in `ratvars`.

  `quotient` returns the first element of the two-element list returned by `divide`.

  Categories: Polynomials ·

Function: rat
rat (expr)
rat (expr, x_1, …, x_n) ¶

  Converts expr to canonical rational expression (CRE) form by expanding and combining all terms over a common denominator and cancelling out the greatest common divisor of the numerator and denominator, as well as converting floating point numbers to rational numbers within a tolerance of `ratepsilon`. The variables are ordered according to the x_1, …, x_n, if specified, as in `ratvars`.

  `rat` does not generally simplify functions other than addition `+`, subtraction `-`, multiplication `*`, division `/`, and exponentiation to an integer power, whereas `ratsimp` does handle those cases. Note that atoms (numbers and variables) in CRE form are not the same as they are in the general form. For example, `rat(x)- x` yields `rat(0)` which has a different internal representation than 0.

  When `ratfac` is `true`, `rat` yields a partially factored form for CRE. During rational operations the expression is maintained as fully factored as possible without an actual call to the factor package. This should always save space and may save some time in some computations. The numerator and denominator are still made relatively prime (e.g., `rat((x^2 - 1)^4/(x + 1)^2)` yields `(x - 1)^4 (x + 1)^2` when `ratfac` is `true`), but the factors within each part may not be relatively prime.

  `ratprint` if `false` suppresses the printout of the message informing the user of the conversion of floating point numbers to rational numbers.

  `keepfloat` if `true` prevents floating point numbers from being converted to rational numbers.

  See also `ratexpand` and `ratsimp`.

  Examples:

      (%i1) ((x - 2*y)^4/(x^2 - 4*y^2)^2 + 1)*(y + a)*(2*y + x) /
            (4*y^2 + x^2);
                                                 4
                                        (x - 2 y)
                    (y + a) (2 y + x) (------------ + 1)
                                         2      2 2
                                       (x  - 4 y )
      (%o1)         ------------------------------------
                                    2    2
                                 4 y  + x

      (%i2) rat (%, y, a, x);
                                  2 a + 2 y
      (%o2)/R/                    ---------
                                   x + 2 y

  Categories: Rational expressions ·

Option variable: ratalgdenom ¶

  Default value: `true`

  When `ratalgdenom` is `true`, allows rationalization of denominators with respect to radicals to take effect. `ratalgdenom` has an effect only when canonical rational expressions (CRE) are used in algebraic mode.

  Categories: Simplification flags and variables ·

Function: ratcoef
ratcoef (expr, x, n)
ratcoef (expr, x) ¶

  Returns the coefficient of the expression `x^n` in the expression expr. If omitted, n is assumed to be 1.

  The return value is free (except possibly in a non-rational sense) of the variables in x. If no coefficient of this type exists, 0 is returned.

  `ratcoef` expands and rationally simplifies its first argument and thus it may produce answers different from those of `coeff` which is purely syntactic. Thus `ratcoef ((x + 1)/y + x, x)` returns `(y + 1)/y` whereas `coeff` returns 1.

  `ratcoef (expr, x, 0)`, viewing expr as a sum, returns a sum of those terms which do not contain x. Therefore if x occurs to any negative powers, `ratcoef` should not be used.

  Since expr is rationally simplified before it is examined, coefficients may not appear quite the way they were envisioned.

  Example:

      (%i1) s: a*x + b*x + 5$

      (%i2) ratcoef (s, a + b);
      (%o2)                           x

  Categories: Polynomials · Rational expressions ·

Function: ratdenom (expr) ¶

  Returns the denominator of expr, after coercing expr to a canonical rational expression (CRE). The return value is a CRE.

  expr is coerced to a CRE by `rat` if it is not already a CRE. This conversion may change the form of expr by putting all terms over a common denominator.

  `denom` is similar, but returns an ordinary expression instead of a CRE. Also, `denom` does not attempt to place all terms over a common denominator, and thus some expressions which are considered ratios by `ratdenom` are not considered ratios by `denom`.

  Categories: Rational expressions ·

Option variable: ratdenomdivide ¶

  Default value: `true`

  When `ratdenomdivide` is `true`, `ratexpand` expands a ratio in which the numerator is a sum into a sum of ratios, all having a common denominator. Otherwise, `ratexpand` collapses a sum of ratios into a single ratio, the numerator of which is the sum of the numerators of each ratio.

  Examples:

      (%i1) expr: (x^2 + x + 1)/(y^2 + 7);
                                  2
                                 x  + x + 1
      (%o1)                      ----------
                                    2
                                   y  + 7

      (%i2) ratdenomdivide: true$

      (%i3) ratexpand (expr);
                             2
                            x        x        1
      (%o3)               ------ + ------ + ------
                           2        2        2
                          y  + 7   y  + 7   y  + 7

      (%i4) ratdenomdivide: false$

      (%i5) ratexpand (expr);
                                  2
                                 x  + x + 1
      (%o5)                      ----------
                                    2
                                   y  + 7

      (%i6) expr2: a^2/(b^2 + 3) + b/(b^2 + 3);
                                           2
                                 b        a
      (%o6)                    ------ + ------
                                2        2
                               b  + 3   b  + 3

      (%i7) ratexpand (expr2);
                                        2
                                   b + a
      (%o7)                        ------
                                    2
                                   b  + 3

  Categories: Simplification flags and variables · Rational expressions ·

Function: ratdiff (expr, x) ¶

  Differentiates the rational expression expr with respect to x. expr must be a ratio of polynomials or a polynomial in x. The argument x may be a variable or a subexpression of expr.

  The result is equivalent to `diff`, although perhaps in a different form. `ratdiff` may be faster than `diff`, for rational expressions.

  `ratdiff` returns a canonical rational expression (CRE) if `expr` is a CRE. Otherwise, `ratdiff` returns a general expression.

  `ratdiff` considers only the dependence of expr on x, and ignores any dependencies established by `depends`.

  Example:

      (%i1) expr: (4*x^3 + 10*x - 11)/(x^5 + 5);
                                 3
                              4 x  + 10 x - 11
      (%o1)                   ----------------
                                    5
                                   x  + 5

      (%i2) ratdiff (expr, x);
                          7       5       4       2
                       8 x  + 40 x  - 55 x  - 60 x  - 50
      (%o2)          - ---------------------------------
                                10       5
                               x   + 10 x  + 25

      (%i3) expr: f(x)^3 - f(x)^2 + 7;
                               3       2
      (%o3)                   f (x) - f (x) + 7

      (%i4) ratdiff (expr, f(x));
                                 2
      (%o4)                   3 f (x) - 2 f(x)

      (%i5) expr: (a + b)^3 + (a + b)^2;
                                    3          2
      (%o5)                  (b + a)  + (b + a)

      (%i6) ratdiff (expr, a + b);
                          2                    2
      (%o6)            3 b  + (6 a + 2) b + 3 a  + 2 a

  Categories: Rational expressions ·

Function: ratdisrep (expr) ¶

  Returns its argument as a general expression. If expr is a general expression, it is returned unchanged.

  Typically `ratdisrep` is called to convert a canonical rational expression (CRE) into a general expression. This is sometimes convenient if one wishes to stop the "contagion", or use rational functions in non-rational contexts.

  See also totaldisrep.

  Categories: Rational expressions ·

Function: ratexpand (expr) ¶
Option variable: ratexpand ¶

  Expands expr by multiplying out products of sums and exponentiated sums, combining fractions over a common denominator, cancelling the greatest common divisor of the numerator and denominator, then splitting the numerator (if a sum) into its respective terms divided by the denominator.

  The return value of `ratexpand` is a general expression, even if expr is a canonical rational expression (CRE).

  The switch `ratexpand` if `true` will cause CRE expressions to be fully expanded when they are converted back to general form or displayed, while if it is `false` then they will be put into a recursive form. See also ratsimp.

  When `ratdenomdivide` is `true`, `ratexpand` expands a ratio in which the numerator is a sum into a sum of ratios, all having a common denominator. Otherwise, `ratexpand` collapses a sum of ratios into a single ratio, the numerator of which is the sum of the numerators of each ratio.

  When `keepfloat` is `true`, prevents floating point numbers from being rationalized when expressions which contain them are converted to canonical rational expression (CRE) form.

  Examples:

      (%i1) ratexpand ((2*x - 3*y)^3);
                           3         2       2        3
      (%o1)          - 27 y  + 54 x y  - 36 x  y + 8 x

      (%i2) expr: (x - 1)/(x + 1)^2 + 1/(x - 1);
                               x - 1       1
      (%o2)                   -------- + -----
                                     2   x - 1
                              (x + 1)

      (%i3) expand (expr);
                          x              1           1
      (%o3)          ------------ - ------------ + -----
                      2              2             x - 1
                     x  + 2 x + 1   x  + 2 x + 1

      (%i4) ratexpand (expr);
                              2
                           2 x                 2
      (%o4)           --------------- + ---------------
                       3    2            3    2
                      x  + x  - x - 1   x  + x  - x - 1

  Categories: Rational expressions ·

Option variable: ratfac ¶

  Default value: `false`

  When `ratfac` is `true`, canonical rational expressions (CRE) are manipulated in a partially factored form.

  During rational operations the expression is maintained as fully factored as possible without calling `factor`. This should always save space and may save time in some computations. The numerator and denominator are made relatively prime, for example `factor ((x^2 - 1)^4/(x + 1)^2)` yields `(x - 1)^4 (x + 1)^2`, but the factors within each part may not be relatively prime.

  In the `ctensor` (Component Tensor Manipulation) package, Ricci, Einstein, Riemann, and Weyl tensors and the scalar curvature are factored automatically when `ratfac` is `true`. `ratfac` should only be set for cases where the tensorial components are known to consist of few terms.

  The `ratfac` and `ratweight` schemes are incompatible and may not both be used at the same time.

  Categories: Rational expressions ·

Function: ratnumer (expr) ¶

  Returns the numerator of expr, after coercing expr to a canonical rational expression (CRE). The return value is a CRE.

  expr is coerced to a CRE by `rat` if it is not already a CRE. This conversion may change the form of expr by putting all terms over a common denominator.

  `num` is similar, but returns an ordinary expression instead of a CRE. Also, `num` does not attempt to place all terms over a common denominator, and thus some expressions which are considered ratios by `ratnumer` are not considered ratios by `num`.

  Categories: Rational expressions ·

Function: ratp (expr) ¶

  Returns `true` if expr is a canonical rational expression (CRE) or extended CRE, otherwise `false`.

  CRE are created by `rat` and related functions. Extended CRE are created by `taylor` and related functions.

  Categories: Predicate functions · Rational expressions ·

Option variable: ratprint ¶

  Default value: `true`

  When `ratprint` is `true`, a message informing the user of the conversion of floating point numbers to rational numbers is displayed.

  Categories: Rational expressions · Numerical evaluation · Console interaction ·

Function: ratsimp (expr) ¶
Function: ratsimp (expr, x_1, …, x_n) ¶

  Simplifies the expression expr and all of its subexpressions, including the arguments to non-rational functions. The result is returned as the quotient of two polynomials in a recursive form, that is, the coefficients of the main variable are polynomials in the other variables. Variables may include non-rational functions (e.g., `sin (x^2 + 1)`) and the arguments to any such functions are also rationally simplified.

  `ratsimp (expr, x_1, ..., x_n)` enables rational simplification with the specification of variable ordering as in `ratvars`.

  When `ratsimpexpons` is `true`, `ratsimp` is applied to the exponents of expressions during simplification.

  See also ratexpand. Note that `ratsimp` is affected by some of the flags which affect `ratexpand`.

  Examples:

      (%i1) sin (x/(x^2 + x)) = exp ((log(x) + 1)^2 - log(x)^2);
                                               2      2
                         x         (log(x) + 1)  - log (x)
      (%o1)        sin(------) = %e
                        2
                       x  + x

      (%i2) ratsimp (%);
                                   1          2
      (%o2)                  sin(-----) = %e x
                                 x + 1

      (%i3) ((x - 1)^(3/2) - (x + 1)*sqrt(x - 1))/sqrt((x - 1)*(x + 1));
                             3/2
                      (x - 1)    - sqrt(x - 1) (x + 1)
      (%o3)           --------------------------------
                           sqrt((x - 1) (x + 1))

      (%i4) ratsimp (%);
                                 2 sqrt(x - 1)
      (%o4)                    - -------------
                                       2
                                 sqrt(x  - 1)

      (%i5) x^(a + 1/a), ratsimpexpons: true;
                                     2
                                    a  + 1
                                    ------
                                      a
      (%o5)                        x

  Categories: Simplification functions · Rational expressions ·

Option variable: ratsimpexpons ¶

  Default value: `false`

  When `ratsimpexpons` is `true`, `ratsimp` is applied to the exponents of expressions during simplification.

  Categories: Simplification flags and variables · Rational expressions ·

Option variable: radsubstflag ¶

  Default value: `false`

  `radsubstflag`, if `true`, permits `ratsubst` to make substitutions such as `u` for `sqrt (x)` in `x`.

  Categories: Simplification flags and variables ·

Function: ratsubst (a, b, c) ¶

  Substitutes a for b in c and returns the resulting expression. b may be a sum, product, power, etc.

  `ratsubst` knows something of the meaning of expressions whereas `subst` does a purely syntactic substitution. Thus `subst (a, x + y, x + y + z)` returns `x + y + z` whereas `ratsubst` returns `z + a`.

  When `radsubstflag` is `true`, `ratsubst` makes substitutions for radicals in expressions which don’t explicitly contain them.

  `ratsubst` ignores the value `true` of the option variables `keepfloat`, `float`, and `numer`.

  Examples:

      (%i1) ratsubst (a, x*y^2, x^4*y^3 + x^4*y^8);
                                    3      4
      (%o1)                      a x  y + a

      (%i2) cos(x)^4 + cos(x)^3 + cos(x)^2 + cos(x) + 1;
                     4         3         2
      (%o2)       cos (x) + cos (x) + cos (x) + cos(x) + 1

      (%i3) ratsubst (1 - sin(x)^2, cos(x)^2, %);
                  4           2                     2
      (%o3)    sin (x) - 3 sin (x) + cos(x) (2 - sin (x)) + 3

      (%i4) ratsubst (1 - cos(x)^2, sin(x)^2, sin(x)^4);
                              4           2
      (%o4)                cos (x) - 2 cos (x) + 1

      (%i5) radsubstflag: false$

      (%i6) ratsubst (u, sqrt(x), x);
      (%o6)                           x

      (%i7) radsubstflag: true$

      (%i8) ratsubst (u, sqrt(x), x);
                                      2
      (%o8)                          u

  Categories: Rational expressions ·

Function: ratvars (x_1, …, x_n) ¶
Function: ratvars () ¶
System variable: ratvars ¶

  Declares main variables x_1, …, x_n for rational expressions. x_n, if present in a rational expression, is considered the main variable. Otherwise, x_[n-1] is considered the main variable if present, and so on through the preceding variables to x_1, which is considered the main variable only if none of the succeeding variables are present.

  If a variable in a rational expression is not present in the `ratvars` list, it is given a lower priority than x_1.

  The arguments to `ratvars` can be either variables or non-rational functions such as `sin(x)`.

  The variable `ratvars` is a list of the arguments of the function `ratvars` when it was called most recently. Each call to the function `ratvars` resets the list. `ratvars ()` clears the list.

  Categories: Rational expressions ·

Option variable: ratvarswitch ¶

  Default value: `true`

  Maxima keeps an internal list in the Lisp variable `VARLIST` of the main variables for rational expressions. If `ratvarswitch` is `true`, every evaluation starts with a fresh list `VARLIST`. This is the default behavior. Otherwise, the main variables from previous evaluations are not removed from the internal list `VARLIST`.

  The main variables, which are declared with the function `ratvars` are not affected by the option variable `ratvarswitch`.

  Examples:

  If `ratvarswitch` is `true`, every evaluation starts with a fresh list `VARLIST`.

      (%i1) ratvarswitch:true$

      (%i2) rat(2*x+y^2);
                                   2
      (%o2)/R/                    y  + 2 x

      (%i3) :lisp varlist
      ($X $Y)

      (%i3) rat(2*a+b^2);
                                   2
      (%o3)/R/                    b  + 2 a

      (%i4) :lisp varlist
      ($A $B)

  If `ratvarswitch` is `false`, the main variables from the last evaluation are still present.

      (%i1) ratvarswitch:false$

      (%i2) rat(2*x+y^2);
                                   2
      (%o2)/R/                    y  + 2 x

      (%i3) :lisp varlist
      ($X $Y)

      (%i3) rat(2*a+b^2);
                                   2
      (%o3)/R/                    b  + 2 a

      (%i4) :lisp varlist
      ($A $B $X $Y)

  Categories: Rational expressions · Global flags ·

Function: ratweight
ratweight (x_1, w_1, …, x_n, w_n)
ratweight () ¶

  Assigns a weight w_i to the variable x_i. This causes a term to be replaced by 0 if its weight exceeds the value of the variable `ratwtlvl` (default yields no truncation). The weight of a term is the sum of the products of the weight of a variable in the term times its power. For example, the weight of `3 x_1^2 x_2` is `2 w_1 + w_2`. Truncation according to `ratwtlvl` is carried out only when multiplying or exponentiating canonical rational expressions (CRE).

  `ratweight ()` returns the cumulative list of weight assignments.

  Note: The `ratfac` and `ratweight` schemes are incompatible and may not both be used at the same time.

  Examples:

      (%i1) ratweight (a, 1, b, 1);
      (%o1)                     [a, 1, b, 1]

      (%i2) expr1: rat(a + b + 1)$

      (%i3) expr1^2;
                        2                  2
      (%o3)/R/         b  + (2 a + 2) b + a  + 2 a + 1

      (%i4) ratwtlvl: 1$

      (%i5) expr1^2;
      (%o5)/R/                  2 b + 2 a + 1

  Categories: Rational expressions ·

System variable: ratweights ¶

  Default value: `[]`

  `ratweights` is the list of weights assigned by `ratweight`. The list is cumulative: each call to `ratweight` places additional items in the list.

  `kill (ratweights)` and `save (ratweights)` both work as expected.

  Categories: Rational expressions ·

Option variable: ratwtlvl ¶

  Default value: `false`

  `ratwtlvl` is used in combination with the `ratweight` function to control the truncation of canonical rational expressions (CRE). For the default value of `false`, no truncation occurs.

  Categories: Rational expressions ·

Function: remainder
remainder (p_1, p_2)
remainder (p_1, p_2, x_1, …, x_n) ¶

  Returns the remainder of the polynomial p_1 divided by the polynomial p_2. The arguments x_1, …, x_n are interpreted as in `ratvars`.

  `remainder` returns the second element of the two-element list returned by `divide`.

  Categories: Polynomials ·

Function: resultant (p_1, p_2, x) ¶

  The function `resultant` computes the resultant of the two polynomials p_1 and p_2, eliminating the variable x. The resultant is a determinant of the coefficients of x in p_1 and p_2, which equals zero if and only if p_1 and p_2 have a non-constant factor in common.

  If p_1 or p_2 can be factored, it may be desirable to call `factor` before calling `resultant`.

  The option variable `resultant` controls which algorithm will be used to compute the resultant. See the option variable `resultant`.

  The function `bezout` takes the same arguments as `resultant` and returns a matrix. The determinant of the return value is the desired resultant.

  Examples:

      (%i1) resultant(2*x^2+3*x+1, 2*x^2+x+1, x);
      (%o1)                           8

      (%i2) resultant(x+1, x+1, x);
      (%o2)                           0

      (%i3) resultant((x+1)*x, (x+1), x);
      (%o3)                           0

      (%i4) resultant(a*x^2+b*x+1, c*x + 2, x);
                               2
      (%o4)                   c  - 2 b c + 4 a

      (%i5) bezout(a*x^2+b*x+1, c*x+2, x);
                              [ 2 a  2 b - c ]
      (%o5)                   [              ]
                              [  c      2    ]

      (%i6) determinant(%);
      (%o6)                   4 a - (2 b - c) c

  Categories: Polynomials ·

Option variable: resultant ¶

  Default value: `subres`

  The option variable `resultant` controls which algorithm will be used to compute the resultant with the function resultant. The possible values are:

  `subres`

    for the subresultant polynomial remainder sequence (PRS) algorithm,

  `mod`

    (not enabled) for the modular resultant algorithm, and

  `red`

    for the reduced polynomial remainder sequence (PRS) algorithm.

  On most problems the default value `subres` should be best.

  Categories: Polynomials ·

Option variable: savefactors ¶

  Default value: `false`

  When `savefactors` is `true`, causes the factors of an expression which is a product of factors to be saved by certain functions in order to speed up later factorizations of expressions containing some of the same factors.

  Categories: Polynomials ·

Function: showratvars (expr) ¶

  Returns a list of the canonical rational expression (CRE) variables in expression `expr`.

  See also ratvars.

  Categories: Rational expressions · Display functions ·

Function: sqfr (expr) ¶

  is similar to `factor` except that the polynomial factors are "square-free." That is, they have factors only of degree one. This algorithm, which is also used by the first stage of `factor`, utilizes the fact that a polynomial has in common with its n’th derivative all its factors of degree greater than n. Thus by taking greatest common divisors with the polynomial of the derivatives with respect to each variable in the polynomial, all factors of degree greater than 1 can be found.

  Example:

      (%i1) sqfr (4*x^4 + 4*x^3 - 3*x^2 - 4*x - 1);
                                      2   2
      (%o1)                  (2 x + 1)  (x  - 1)

  Categories: Polynomials ·

Function: tellrat
tellrat (p_1, …, p_n)
tellrat () ¶

  Adds to the ring of algebraic integers known to Maxima the elements which are the solutions of the polynomials p_1, …, p_n. Each argument p_i is a polynomial with integer coefficients.

  `tellrat (x)` effectively means substitute 0 for x in rational functions.

  `tellrat ()` returns a list of the current substitutions.

  `algebraic` must be set to `true` in order for the simplification of algebraic integers to take effect.

  Maxima initially knows about the imaginary unit `%i` and all roots of integers.

  There is a command `untellrat` which takes kernels and removes `tellrat` properties.

  When `tellrat`’ing a multivariate polynomial, e.g., `tellrat (x^2 - y^2)`, there would be an ambiguity as to whether to substitute `y^2` for `x^2` or vice versa. Maxima picks a particular ordering, but if the user wants to specify which, e.g. `tellrat (y^2 = x^2)` provides a syntax which says replace `y^2` by `x^2`.

  Examples:

      (%i1) 10*(%i + 1)/(%i + 3^(1/3));
                                 10 (%i + 1)
      (%o1)                      -----------
                                        1/3
                                  %i + 3

      (%i2) ev (ratdisrep (rat(%)), algebraic);
                   2/3      1/3              2/3      1/3
      (%o2)    (4 3    - 2 3    - 4) %i + 2 3    + 4 3    - 2

      (%i3) tellrat (1 + a + a^2);
                                  2
      (%o3)                     [a  + a + 1]

      (%i4) 1/(a*sqrt(2) - 1) + a/(sqrt(3) + sqrt(2));
                            1                 a
      (%o4)           ------------- + -----------------
                      sqrt(2) a - 1   sqrt(3) + sqrt(2)

      (%i5) ev (ratdisrep (rat(%)), algebraic);
                                    3/2           3/2
                    (7 sqrt(3) - 5 2    + 2) a - 2    - 1
      (%o5)         -------------------------------------
                                      7

      (%i6) tellrat (y^2 = x^2);
                              2    2   2
      (%o6)                 [y  - x , a  + a + 1]

  Categories: Polynomials · Rational expressions ·

Function: totaldisrep (expr) ¶

  Converts every subexpression of expr from canonical rational expressions (CRE) to general form and returns the result. If expr is itself in CRE form then `totaldisrep` is identical to `ratdisrep`.

  `totaldisrep` may be useful for ratdisrepping expressions such as equations, lists, matrices, etc., which have some subexpressions in CRE form.

  Categories: Rational expressions ·

Function: untellrat (x_1, …, x_n) ¶

  Removes `tellrat` properties from x_1, …, x_n.

  Categories: Polynomials · Rational expressions ·

### 16.3 Introduction to algebraic extensions ¶

We assume here that the fields are of characteristic 0 so that irreductible polynomials have simple roots (are separable, thus square free). The base fields \(K\) of interest are the field \(Q\) of rational numbers, for algebraic numbers, and the fields of rational functions on the real numbers \(R\) or the complex numbers \(C\), that is \(R(t)\) or \(C(t)\), when considering algebraic functions. An extension of degree \(n\) is defined by an irreducible degree \(n\) polynomial \(p(x)\) with coefficients in the base field, and consists of the quotient of the ring \(K[x]\) of polynomials by the multiples of \(p(x)\). So if \(p(x) = x^n + p_0 x^{n - 1} + ... + p_n\), each time one encounters \(x^n\) one substitutes \(-(p_0 x^{n - 1} + ... + p_n)\). This is a field because of Bezout’s identity, and a vector space of dimension \(n\) over \(K\) spanned by \(1, x, ..., x^{n - 1}\). When \(K = C(t)\), this field can be identified with the field of algebraic functions on the algebraic curve of equation \(p(x, t) = 0\).

In Maxima the process of taking rationals modulo \(p\) is obtained by the function `tellrat` when `algebraic` is true. The best way to ensure, in particular when considering the case where \(p\) depends on other variables that this simplification property is attached to \(x\) is to write (note the polynomial must be monic): `tellrat(x^n = -(p_0*x^(n - 1) + ... + p_n))` where the \(p_i\) may depend on other variables. When one wants to remove this tellrat property one then has to write `untellrat(x)`.

In the field \(K[x]\) one may do all sorts of algebraic computations, taking quotients, GCD of two elements, etc. by the same algorithms as in the usual case. In particular one can do factorization of polynomials on an extension, using the function `algfac` below. Moreover multiplication by an element \(f\) is a linear operation of the vector space \(K[x]\) over \(K\) and as such has a trace and a determinant. These are called `algtrace` and `algnorm` below. One can see that the trace of an element \(f(x)\) in \(K[x]\) is the sum of the values \(f(a)\) when \(a\) runs over roots of \(p\) and the norm is the product of the \(f(a)\). Both are symmetric in the roots of \(p\) and thus belong to \(K\).

The field \(K[x]\) is also called the field obtained by adjoining a root \(a\) of \(p(x)\) to \(K\). One can similarly adjoin a second root \(b\) of another polynomial obtaining a new extension \(K[a,b]\). In fact there is a “prime element” \(c\) in \(K[a, b]\) such that \(K[a, b] = K[c]\). This is obtained by function `primeelmt` below. Recursively one can thus adjoin any number of elements. In particular adjoining all the roots of \(p(x)\) to \(K\) one gets the splitting field of \(p\), which is the smallest extension in which \(p\) completely splits in linear functions. The function `splitfield` constructs a primitive element of the splitting field, which in general is of very high degree.

The relevant concepts are explained in a concise and self-contained way in the small books edited by Dover: “Algebraic theory of numbers,” by Pierre Samuel, “Algebraic curves,” by Robert Walker, and the methods presented here are described in the article “Algebraic factoring and rational function integration” by B. Trager, Proceedings of the 1976 AMS Symposium on Symbolic and Algebraic Computation.

### 16.4 Functions and Variables for algebraic extensions ¶

Function: algfac (f, p) ¶

  Returns the factorization of f in the field \(K[a]\). Does the same as `factor(f, p)` which in fact calls `algfac`. One can also specify the variable a as in `algfac(f, p, a)`.

  Examples:

      (%i1) algfac(x^4 + 1, a^2 - 2);
                                 2              2
      (%o1)                    (x  - a x + 1) (x  + a x + 1)
      (%i2) algfac(x^4 - t*x^2 + 1, a^2 - t - 2, a);
                                 2              2
      (%o2)                    (x  - a x + 1) (x  + a x + 1)

  In the second example note that \(a = sqrt(2 + t)\).

Function: algnorm (f, p, a) ¶

  Returns the norm of the polynomial \(f(a)\) in the extension obtained by a root a of polynomial p. The coefficients of f may depend on other variables.

  Examples:

      (%i1) algnorm(x*a^2 + y*a + z,a^2 - 2, a);
                                  2              2      2
      (%o1)/R/                   z  + 4 x z - 2 y  + 4 x

  The norm is also the resultant of polynomials f and p, and the product of the differences of the roots of f and p.

Function: algtrace (f, p, a) ¶

  Returns the trace of the polynomial \(f(a)\) in the extension obtained by a root a of polynomial p. The coefficients of f may depend on other variables which remain “inert”.

  Example:

      (%i1) algtrace(x*a^5 + y*a^3 + z + 1, a^2 + a + 1, a);
      (%o1)/R/                       2 z + 2 y - x + 2

Function: bdiscr (args) ¶

  Computes the discriminant of a basis \(x_i\) in \(K[a]\) as the determinant of the matrix of elements \(trace(x_i*x_j)\). The args are the elements of the basis followed by the minimal polynomial.

  Example:

      (%i1) bdiscr(1, x, x^2, x^3 - 2);
      (%o1)/R/                             - 108
      (%i2) poly_discriminant(x^3 - 2, x);
      (%o2)                                - 108

  A standard base in an extension of degree n is \(1, x, ..., x^{n - 1}\). In this case it is known that the discriminant of this base is the discriminant of the minimal polynomial. This is checked in (%o2) above.

Function: primelmt (f_b, p_a, c) ¶

  Computes a prime element for the extension of \(K[a]\) by a root b of a polynomial \(f_b(b)\) whose coefficients may depend on a. One assumes that f_b is square free. The function returns an irreducible polynomial, a root of which generates \(K[a, b]\), and the expression of this primitive element in terms of a and b.

  Examples:

      (%i1) primelmt(b^2 - a*b - 1, a^2 - 2, c);
                                    4       2
      (%o1)                       [c  - 12 c  + 9, b + a]
      (%i2) solve(b^2 - sqrt(2)*b - 1)[1];
                                        sqrt(6) - sqrt(2)
      (%o2)                       b = - -----------------
                                                2
      (%i3) primelmt(b^2 - 3, a^2 - 2, c);
                                    4       2
      (%o3)                       [c  - 10 c  + 1, b + a]
      (%i4) factor(c^4 - 12*c^2 + 9, a^4 - 10*a^2 + 1);
                       3    2                       3    2
      (%o4) ((4 c - 3 a  - a  + 27 a + 5) (4 c - 3 a  + a  + 27 a - 5)
                                 3    2                       3    2
                       (4 c + 3 a  - a  - 27 a + 5) (4 c + 3 a  + a  - 27 a - 5))/256
      (%i5) primelmt(b^3 - 3, a^2 - 2, c);
                         6      4      3       2
      (%o5)            [c  - 6 c  - 6 c  + 12 c  - 36 c + 1, b + a]
      (%i6) factor(b^3 - 3, %[1]);
                  5       4        3        2
      (%o6) ((48 c  + 27 c  - 320 c  - 468 c  + 124 c + 755 b - 1092)
                 5        5         4       4          3        3          2        2
       ((- 48 b c ) - 54 c  - 27 b c  + 64 c  + 320 b c  + 360 c  + 468 b c  + 149 c
                                 2
       - 124 b c - 1272 c + 755 b  + 1092 b + 1606))/570025

  In (%o1), f_b depends on `a`. Using `solve`, the solution depends on sqrt(2) and sqrt(3). In (%o3), \(K[sqrt(2), sqrt(3)]\) is computed, and we see that the the primitive polynomial in (%o1) factorizes completely here. In (%i5), we compute \(K[sqrt(2), 3^{1/3}]\), and we see that `b^3 - 3` gets one factor in this extension. If we assume this extension is real, the two other factors are complex.

Function: splitfield (p, x) ¶

  Computes the splitting field of the polynomial \(p(x)\). In the generic case it is of degree \(n!\) in terms of the degree \(n\) of p, but may be of lower order if the Galois group of p is a strict subgroup of the group of permutations of \(n\) elements. The function returns a primitive polynomial for this extension and the expressions of the roots of p as polynomials of a root of this primitive polynomial. The polynomial f may be irreducible or factorizable.

  Examples:

      (%i1) splitfield(x^3 + x + 1, x);
                                                    4         2
                    6         4         2       alg1  + 5 alg1  - 9 alg1 + 4
      (%o1)/R/ [alg1  + 6 alg1  + 9 alg1  + 31, ----------------------------,
                                                             18
                                       4         2          4         2
                                   alg1  + 5 alg1  + 4  alg1  + 5 alg1  + 9 alg1 + 4
                                 - -------------------, ----------------------------]
                                            9                        18
      (%i2) splitfield(x^4 + 10*x^2 - 96*x - 71, x)[1];
                   8           6           5            4             3
      (%o2)/R/ alg2  + 148 alg2  - 576 alg2  + 9814 alg2  - 42624 alg2
                                                          2
                                             + 502260 alg2  + 1109952 alg2 + 18860337

  In the first case we have the primitive polynomial of degree 6 and the 3 roots of the third degree equations in terms of a variable `alg1` produced by the system. In the second case the primitive polynomial is of degree 8 instead of 24, because the Galois group of the equation is reduced to D8 since there are relations between the roots.

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