content

Content of a polynomial

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Syntax

content(p)
content(f, <vars>)

Description

content(p) computes the content of the polynomial p or polynomial expression, i.e., the greatest common divisor of its coefficients.

If p is the zero polynomial, then content returns 0.

If p is a non-zero polynomial with coefficient ring IntMod(n) and n is a prime number, then content returns 1. If n is not a prime number, an error message is issued.

If p is a polynomial with a library domain R as coefficient ring, the gcd of its coefficients is computed using the slot gcd of R. If no such slot exists, then content returns FAIL.

If p is a polynomial with coefficient ring Expr, then content does the following.

If all coefficients of p are either integers or rational numbers, content(p) is equivalent to gcd(coeff(p)), and the return value is a positive integer or rational number. See Example 1.

If at least one coefficient is a floating point number or a complex number and all other coefficients are numbers, then content returns 1. See Example 2.

If at least one coefficient is not a number and all coefficients of p can be converted into polynomials via poly, then content(p) is equivalent to gcd(coeff(p)). See Example 3.

Otherwise, content returns 1.

A polynomial expression f is converted into a polynomial with coefficient ring Expr via p :=poly(f, vars), and then content is applied to p. See Example 1.

Use icontent for polynomials that are known to have integer or rational coefficients, since it is much faster than content.

Dividing the coefficients of p by its content gives its primitive part. This one can also be obtained directly using polylib::primpart.

Examples

Example 1

If p is a polynomial with integer or rational coefficients, the result is the same as for icontent:

content(poly(6*x^3*y + 3*x*y + 9*y, [x, y]))

The following call, where the first argument is a polynomial expression and not a polynomial, is equivalent to the one above:

content(6*x^3*y + 3*x*y + 9*y, [x, y])

If no list of indeterminates is specified, then poly converts the expression into a polynomial with respect to all occurring indeterminates, and we obtain yet another equivalent call:

content(6*x^3*y + 3*x*y + 9*y)

Above, we considered the polynomial as a bivariate polynomial with integer coefficients. We can also consider the same expression as a univariate polynomial in x, whose coefficients contain a parameter y. Then the coefficients and their gcd—the content—are polynomial expressions in y:

content(poly(6*x^3*y + 3*x*y + 9*y, [x]))

Here is another example where the coefficients and the content are again polynomial expressions:

content(poly(4*x*y + 6*x^3 + 6*x*y^2 + 9*x^3*y, [x]))

The following call is equivalent to the previous one:

content(4*x*y + 6*x^3 + 6*x*y^2 + 9*x^3*y, [x])

Example 2

If a polynomial or polynomial expression has numeric coefficients and at least one floating-point number is among them, its content is 1:

content(2.0*x+2.0)

Example 3

If not all of the coefficients are numbers, the gcd of the coefficients is returned:

content(poly(x^2*y+x, [y]))

Parameters

p

A polynomial of type DOM_POLY

f

A polynomial expression

vars

A list of indeterminates of the polynomial: typically, identifiers or indexed identifiers

Return Values

an object of the same type as the coefficients of the polynomial or the value FAIL.

Overloaded By

p

See Also

MuPAD Functions