RegularChains[ChainTools] - Maple Programming Help

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RegularChains[ChainTools]

 SquarefreeFactorization
 compute a squarefree decomposition of a polynomial modulo a regular chain

 Calling Sequence SquarefreeFactorization(p, v, rc, R) SquarefreeFactorization(p, v, rc, R,options)

Parameters

 p - polynomial v - variable rc - regular chain R - polynomial ring options - equation of the form 'method'=mth, where mth is either 'evala' or 'src'

Description

 • The command SquarefreeFactorization(p, v, rc, R) returns a list of pairs [sqf_i, rc_i].  For each pair, the list sqf_i is a squarefree decomposition of p modulo the saturated ideal of rc_i; each element in the list sqf_i is a pair as [s_j,e_j], where s_j is a squarefree polynomial modulo rc_i and e_j is the exponent of s_j in p.
 • All the regular chains from the output pairs form a triangular decomposition of rc in the sense of Kalkbrener.
 • The option 'method' specifies which gcd algorithm to use. The default option, 'method'='evala', uses a modular algorithm. The other option is 'method'='src', which uses a subresultant-based approach. This method is generally slower, but can be faster in some cases, for instance, if the dimension of the saturated ideal of rc is high, say greater than 4.
 • Assumptions: the polynomial ring is assumed to have characteristic zero; the initial of p is regular w.r.t. rc; v is greater than the main variables of the regular chain rc in R.

Examples

Example 1

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$$\mathrm{with}\left(\mathrm{ChainTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[y,x\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)
 > $q≔\mathrm{expand}\left({\left(x-2\right)}^{2}{\left(x-1\right)}^{2}\right)$
 ${q}{≔}{{x}}^{{4}}{-}{6}{}{{x}}^{{3}}{+}{13}{}{{x}}^{{2}}{-}{12}{}{x}{+}{4}$ (2)
 > $p≔\mathrm{prem}\left(\left(y+6\right)\left(y+2-x\right)\left(x{y}^{2}-1\right)\left(y-x\right),q,x\right)$
 ${p}{≔}{{x}}^{{3}}{}{{y}}^{{3}}{-}{2}{}{{x}}^{{2}}{}{{y}}^{{4}}{+}{x}{}{{y}}^{{5}}{+}{6}{}{{x}}^{{3}}{}{{y}}^{{2}}{-}{14}{}{{x}}^{{2}}{}{{y}}^{{3}}{+}{8}{}{x}{}{{y}}^{{4}}{-}{12}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{12}{}{x}{}{{y}}^{{3}}{-}{{x}}^{{2}}{}{y}{+}{2}{}{x}{}{{y}}^{{2}}{-}{{y}}^{{3}}{-}{6}{}{{x}}^{{2}}{+}{14}{}{y}{}{x}{-}{8}{}{{y}}^{{2}}{+}{12}{}{x}{-}{12}{}{y}$ (3)
 > $C≔\mathrm{Chain}\left(\left[q\right],\mathrm{Empty}\left(R\right),R\right)$
 ${C}{≔}{\mathrm{regular_chain}}$ (4)
 > $\mathrm{SquarefreeFactorization}\left(p,y,C,R\right)$
 $\left[\left[\left[\left[{2}{}{{y}}^{{5}}{+}{8}{}{{y}}^{{4}}{-}{25}{}{{y}}^{{3}}{-}{4}{}{{y}}^{{2}}{+}{12}{}{y}{,}{1}\right]\right]{,}{\mathrm{regular_chain}}\right]{,}\left[\left[\left[{y}{+}{6}{,}{1}\right]{,}\left[{{y}}^{{2}}{-}{1}{,}{2}\right]\right]{,}{\mathrm{regular_chain}}\right]\right]$ (5)
 > $\mathrm{SquarefreeFactorization}\left(p,y,C,R,\mathrm{method}='\mathrm{src}'\right)$
 $\left[\left[\left[\left[{2}{}{{y}}^{{5}}{+}{8}{}{{y}}^{{4}}{-}{25}{}{{y}}^{{3}}{-}{4}{}{{y}}^{{2}}{+}{12}{}{y}{,}{1}\right]\right]{,}{\mathrm{regular_chain}}\right]{,}\left[\left[\left[{y}{+}{6}{,}{1}\right]{,}\left[{{y}}^{{2}}{-}{1}{,}{2}\right]\right]{,}{\mathrm{regular_chain}}\right]\right]$ (6)

Example 2

 > $R≔\mathrm{PolynomialRing}\left(\left[x,y\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (7)
 > $p≔\left(x+y\right){\left(x+2y\right)}^{2}{\left(x+3y\right)}^{3}$
 ${p}{≔}\left({x}{+}{y}\right){}{\left({x}{+}{2}{}{y}\right)}^{{2}}{}{\left({x}{+}{3}{}{y}\right)}^{{3}}$ (8)
 > $\mathrm{rc}≔\mathrm{Chain}\left(\left[{y}^{2}+1\right],\mathrm{Empty}\left(R\right),R\right)$
 ${\mathrm{rc}}{≔}{\mathrm{regular_chain}}$ (9)
 > $p≔\mathrm{SparsePseudoRemainder}\left(p,\mathrm{rc},R\right)$
 ${p}{≔}{{x}}^{{6}}{+}{14}{}{{x}}^{{5}}{}{y}{-}{80}{}{{x}}^{{4}}{-}{238}{}{{x}}^{{3}}{}{y}{+}{387}{}{{x}}^{{2}}{+}{324}{}{y}{}{x}{-}{108}$ (10)
 > $\mathrm{SquarefreeFactorization}\left(p,x,\mathrm{rc},R\right)$
 $\left[\left[\left[\left[{x}{+}{y}{,}{1}\right]{,}\left[{x}{+}{2}{}{y}{,}{2}\right]{,}\left[{x}{+}{3}{}{y}{,}{3}\right]\right]{,}{\mathrm{rc}}\right]\right]$ (11)

Example 3

 > $\mathrm{rc}≔\mathrm{Chain}\left(\left[{y}^{2}+y\right],\mathrm{Empty}\left(R\right),R\right)$
 ${\mathrm{rc}}{≔}{\mathrm{regular_chain}}$ (12)
 > $p≔\left(x+y\right){\left(x+2y\right)}^{2}{\left(x+3y\right)}^{3}$
 ${p}{≔}\left({x}{+}{y}\right){}{\left({x}{+}{2}{}{y}\right)}^{{2}}{}{\left({x}{+}{3}{}{y}\right)}^{{3}}$ (13)
 > $p≔\mathrm{SparsePseudoRemainder}\left(p,\mathrm{rc},R\right)$
 ${p}{≔}{{x}}^{{6}}{+}{14}{}{{x}}^{{5}}{}{y}{-}{80}{}{{x}}^{{4}}{}{y}{+}{238}{}{{x}}^{{3}}{}{y}{-}{387}{}{{x}}^{{2}}{}{y}{+}{324}{}{y}{}{x}{-}{108}{}{y}$ (14)
 > $\mathrm{SquarefreeFactorization}\left(p,x,\mathrm{rc},R\right)$
 $\left[\left[\left[\left[{x}{-}{1}{,}{1}\right]{,}\left[{x}{-}{2}{,}{2}\right]{,}\left[{x}{-}{3}{,}{3}\right]\right]{,}{\mathrm{regular_chain}}\right]{,}\left[\left[\left[{x}{,}{6}\right]\right]{,}{\mathrm{regular_chain}}\right]\right]$ (15)
 > $\mathrm{SquarefreeFactorization}\left(p,x,\mathrm{rc},R,\mathrm{method}='\mathrm{src}'\right)$
 $\left[\left[\left[\left[{x}{-}{1}{,}{1}\right]{,}\left[{x}{-}{2}{,}{2}\right]{,}\left[{x}{-}{3}{,}{3}\right]\right]{,}{\mathrm{regular_chain}}\right]{,}\left[\left[\left[{x}{,}{6}\right]\right]{,}{\mathrm{regular_chain}}\right]\right]$ (16)

Compatibility

 • The RegularChains[ChainTools][SquarefreeFactorization] command was introduced in Maple 16.