Factors - Maple Programming Help

Factors

inert factors function

 Calling Sequence Factors(a, K)

Parameters

 a - multivariate polynomial K - optional specification for an algebraic extension

Description

 • The Factors function is a placeholder for representing the factorization of the multivariate polynomial a over U, a unique factorization domain. The construct Factors(a) produces a data structure of the form $[u,[[{f}_{1},{ⅇ}_{1}],...,[{f}_{n},{ⅇ}_{n}]]]$ such that $a=u{{f}_{1}}^{{ⅇ}_{1}}...{{f}_{n}}^{{ⅇ}_{n}}$, where each f[i] is a primitive irreducible polynomial.
 • The difference between the Factors function and the Factor function is only the form of the result.  The Factor function, if defined, returns a Maple sum of products more suitable for interactive display and manipulation.
 • The call Factors(a) mod p computes the factorization of a over the integers modulo p, a prime integer. The polynomial a must have rational coefficients or coefficients over a finite field specified by RootOfs.
 • The call Factors(a, K) mod p computes the factorization over the finite field defined by K, an algebraic extension of the integers mod p where K is a RootOf.
 • The call modp1(Factors(a),p) computes the factorization of the polynomial a in the $\mathrm{modp1}$ representation modulo p a prime integer.
 • The call evala(Factors(a, K)) computes the factorization of the polynomial a over an algebraic number (or function) field defined by the extension K, which is specified as a RootOf or a set of RootOfs. The polynomial a must have algebraic number (or function) coefficients. The factors are monic for the ordering of the variables chosen by Maple.

Examples

 > $\mathrm{Factors}\left(2{x}^{2}+6x+6\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}7$
 $\left[{2}{,}\left[\left[{x}{+}{4}{,}{1}\right]{,}\left[{x}{+}{6}{,}{1}\right]\right]\right]$ (1)
 > $\mathrm{Factors}\left({x}^{5}+1\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}2$
 $\left[{1}{,}\left[\left[{{x}}^{{4}}{+}{{x}}^{{3}}{+}{{x}}^{{2}}{+}{x}{+}{1}{,}{1}\right]{,}\left[{x}{+}{1}{,}{1}\right]\right]\right]$ (2)
 > $\mathrm{alias}\left(\mathrm{α}=\mathrm{RootOf}\left({x}^{2}+x+1\right)\right)$
 ${\mathrm{α}}$ (3)
 > $\mathrm{Factors}\left({x}^{5}+1,\mathrm{α}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}2$
 $\left[{1}{,}\left[\left[{x}{+}{1}{,}{1}\right]{,}\left[{\mathrm{α}}{}{x}{+}{{x}}^{{2}}{+}{x}{+}{1}{,}{1}\right]{,}\left[{\mathrm{α}}{}{x}{+}{{x}}^{{2}}{+}{1}{,}{1}\right]\right]\right]$ (4)
 > $\mathrm{alias}\left(\mathrm{sqrt2}=\mathrm{RootOf}\left({x}^{2}-2\right)\right):$
 > $\mathrm{evala}\left(\mathrm{Factors}\left(2{x}^{2}-1,\mathrm{sqrt2}\right)\right)$
 $\left[{2}{,}\left[\left[{x}{+}\frac{{1}}{{2}}{}{\mathrm{sqrt2}}{,}{1}\right]{,}\left[{x}{-}\frac{{1}}{{2}}{}{\mathrm{sqrt2}}{,}{1}\right]\right]\right]$ (5)
 > $\mathrm{alias}\left(\mathrm{sqrtx}=\mathrm{RootOf}\left({y}^{2}-x,y\right)\right):$
 > $\mathrm{evala}\left(\mathrm{Factors}\left(x{y}^{2}-1,\mathrm{sqrtx}\right)\right)$
 $\left[{x}{,}\left[\left[{y}{+}\frac{{\mathrm{sqrtx}}}{{x}}{,}{1}\right]{,}\left[{y}{-}\frac{{\mathrm{sqrtx}}}{{x}}{,}{1}\right]\right]\right]$ (6)
 > $\mathrm{expand}\left(\left({x}^{3}+{y}^{5}+2\right)\left(x{y}^{2}+3\right)\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}7$
 ${x}{}{{y}}^{{7}}{+}{{x}}^{{4}}{}{{y}}^{{2}}{+}{3}{}{{y}}^{{5}}{+}{3}{}{{x}}^{{3}}{+}{2}{}{x}{}{{y}}^{{2}}{+}{6}$ (7)
 > $\mathrm{Factors}\left(\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}7$
 $\left[{1}{,}\left[\left[{{y}}^{{5}}{+}{{x}}^{{3}}{+}{2}{,}{1}\right]{,}\left[{x}{}{{y}}^{{2}}{+}{3}{,}{1}\right]\right]\right]$ (8)
 > $\mathrm{Factors}\left({x}^{2}+2xy+{y}^{2}+1+x+y,\mathrm{α}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}5$
 $\left[{1}{,}\left[\left[{y}{+}{x}{+}{\mathrm{α}}{+}{1}{,}{1}\right]{,}\left[{y}{+}{x}{+}{4}{}{\mathrm{α}}{,}{1}\right]\right]\right]$ (9)
 > $\mathrm{Factors}\left({x}^{2}y+x{y}^{2}+2\mathrm{α}xy+\mathrm{α}{x}^{2}+4\mathrm{α}x+y+\mathrm{α}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}5$
 $\left[{1}{,}\left[\left[{y}{+}{x}{+}{\mathrm{α}}{,}{1}\right]{,}\left[{\mathrm{α}}{}{x}{+}{x}{}{y}{+}{1}{,}{1}\right]\right]\right]$ (10)
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