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The function resultant computes the resultant of the two polynomials a and b with respect to the indeterminate x.

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If a and b are polynomials over an integral domain, where

$a={a}_{n}\left(\prod _{i=1}^{n}\left(x{\mathrm{\alpha}}_{i}\right)\right)$
$b={b}_{m}\left(\prod _{i=1}^{m}\left(x{\mathrm{\beta}}_{i}\right)\right)$

then the resultant of the two polynomials a and b with respect to x is defined to be the product

${a}_{n}^{m}{b}_{m}^{n}\left(\prod _{i=1}^{n}\phantom{\rule[0.0ex]{0.4em}{0.0ex}}\prod _{j=1}^{m}\left({\mathrm{\alpha}}_{i}{\mathrm{\beta}}_{j}\right)\right)$
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The resultant can be computed from the Euclidean algorithm, or computed as the determinant of Sylvester's matrix or Bezout's matrix. For univariate and bivariate resultants over the rationals, modular methods are used for polynomials of high degree and the subresultant algorithm is used for polynomials of low degree. Otherwise Bezout's determinant is computed using minor expansion.

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For univariate or bivariate problems containing integer entries, the modular method produces a probabilistic result. Information on controlling the probabilistic behavior can be found in EnvProbabilistic.

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For efficient computation, resultant takes advantage of any factorization of a and b that is present, although no explicit factorization is attempted.
