10.6.3 Decomposing Polynomials

The poly function takes a polynomial and a variable as arguments, and returns a vector of polynomial coefficients (constant coefficient first). For example, ‘poly(x^3 + 2 x, x)’ returns ‘[0, 2, 0, 1]’. If the input is not a polynomial in ‘x’, the call to poly is left in symbolic form. If the input does not involve the variable ‘x’, the input is returned in a list of length one, representing a polynomial with only a constant coefficient. The call ‘poly(x, x)’ returns the vector ‘[0, 1]’. The last element of the returned vector is guaranteed to be nonzero; note that ‘poly(0, x)’ returns the empty vector ‘[]’. Note also that ‘x’ may actually be any formula; for example, ‘poly(sin(x)^2 - sin(x) + 3, sin(x))’ returns ‘[3, -1, 1]’.

To get the ‘x^k’ coefficient of polynomial ‘p’, use ‘poly(p, x)_(k+1)’. To get the degree of polynomial ‘p’, use ‘vlen(poly(p, x)) - 1’. For example, ‘poly((x+1)^4, x)’ returns ‘[1, 4, 6, 4, 1]’, so ‘poly((x+1)^4, x)_(2+1)’ gives the ‘x^2’ coefficient of this polynomial, 6.

One important feature of the solver is its ability to recognize formulas which are “essentially” polynomials. This ability is made available to the user through the gpoly function, which is used just like poly: ‘gpoly(expr, var)’. If expr is a polynomial in some term which includes var, then this function will return a vector ‘[x, c, a]’ where x is the term that depends on var, c is a vector of polynomial coefficients (like the one returned by poly), and a is a multiplier which is usually 1. Basically, ‘expr = a*(c_1 + c_2 x + c_3 x^2 + ...)’. The last element of c is guaranteed to be non-zero, and c will not equal ‘[1]’ (i.e., the trivial decomposition expr = x is not considered a polynomial). One side effect is that ‘gpoly(x, x)’ and ‘gpoly(6, x)’, both of which might be expected to recognize their arguments as polynomials, will not because the decomposition is considered trivial.

For example, ‘gpoly((x-2)^2, x)’ returns ‘[x, [4, -4, 1], 1]’, since the expanded form of this polynomial is ‘4 - 4 x + x^2’.

The term x may itself be a polynomial in var. This is done to reduce the size of the c vector. For example, ‘gpoly(x^4 + x^2 - 1, x)’ returns ‘[x^2, [-1, 1, 1], 1]’, since a quadratic polynomial in ‘x^2’ is easier to solve than a quartic polynomial in ‘x’.

A few more examples of the kinds of polynomials gpoly can discover:

sin(x) - 1               [sin(x), [-1, 1], 1]
x + 1/x - 1              [x, [1, -1, 1], 1/x]
x + 1/x                  [x^2, [1, 1], 1/x]
x^3 + 2 x                [x^2, [2, 1], x]
x + x^2:3 + sqrt(x)      [x^1:6, [1, 1, 0, 1], x^1:2]
x^(2a) + 2 x^a + 5       [x^a, [5, 2, 1], 1]
(exp(-x) + exp(x)) / 2   [e^(2 x), [0.5, 0.5], e^-x]

The poly and gpoly functions accept a third integer argument which specifies the largest degree of polynomial that is acceptable. If this is ‘n’, then only c vectors of length ‘n+1’ or less will be returned. Otherwise, the poly or gpoly call will remain in symbolic form. For example, the equation solver can handle quartics and smaller polynomials, so it calls ‘gpoly(expr, var, 4)’ to discover whether expr can be treated by its linear, quadratic, cubic, or quartic formulas.

The pdeg function computes the degree of a polynomial; ‘pdeg(p,x)’ is the highest power of x that appears in p. This is the same as ‘vlen(poly(p,x))-1’, but is much more efficient. If p is constant with respect to x, then ‘pdeg(p,x) = 0’. If p is not a polynomial in x (e.g., ‘pdeg(2 cos(x), x)’, the function remains unevaluated. It is possible to omit the second argument x, in which case ‘pdeg(p)’ returns the highest total degree of any term of the polynomial, counting all variables that appear in p. Note that pdeg(c) = pdeg(c,x) = 0 for any nonzero constant c; the degree of the constant zero is considered to be -inf (minus infinity).

The plead function finds the leading term of a polynomial. Thus ‘plead(p,x)’ is equivalent to ‘poly(p,x)_vlen(poly(p,x))’, though again more efficient. In particular, ‘plead((2x+1)^10, x)’ returns 1024 without expanding out the list of coefficients. The value of plead(p,x) will be zero only if ‘p = 0’.

The pcont function finds the content of a polynomial. This is the greatest common divisor of all the coefficients of the polynomial. With two arguments, pcont(p,x) effectively uses ‘poly(p,x)’ to get a list of coefficients, then uses pgcd (the polynomial GCD function) to combine these into an answer. For example, ‘pcont(4 x y^2 + 6 x^2 y, x)’ is ‘2 y’. The content is basically the “biggest” polynomial that can be divided into p exactly. The sign of the content is the same as the sign of the leading coefficient.

With only one argument, ‘pcont(p)’ computes the numerical content of the polynomial, i.e., the gcd of the numerical coefficients of all the terms in the formula. Note that gcd is defined on rational numbers as well as integers; it computes the gcd of the numerators and the lcm of the denominators. Thus ‘pcont(4:3 x y^2 + 6 x^2 y)’ returns 2:3. Dividing the polynomial by this number will clear all the denominators, as well as dividing by any common content in the numerators. The numerical content of a polynomial is negative only if all the coefficients in the polynomial are negative.

The pprim function finds the primitive part of a polynomial, which is simply the polynomial divided (using pdiv if necessary) by its content. If the input polynomial has rational coefficients, the result will have integer coefficients in simplest terms.