RegularChains[ParametricSystemTools] - Maple Programming Help

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RegularChains[ParametricSystemTools]

 CoefficientsInParameters
 return the coefficients of a polynomial with respect to parameters

 Calling Sequence CoefficientsInParameters(p, d, R)

Parameters

 p - polynomial in the ring d - number of parameters R - polynomial ring

Description

 • The command CoefficientsInParameters(p, d, R) returns a list, lp, of polynomials involving the last d variables only.
 • The integer d should be positive and less than the number of variables. The last d variables are regarded as parameters, ${U}_{1},...,{U}_{d}$, and the other variables, ${X}_{1},...,{X}_{n}$, are regarded as unknowns.
 • The common zeros of the polynomials in $\mathrm{lp}$ form the variety of ${K}^{d}$ where the polynomial $p$ is identically zero when regarded as a polynomial in ${X}_{1},...,{X}_{n}$ with coefficients in $K[{U}_{1},...,{U}_{d}]$ where $K$ is the algebraic closure of the ground field of R.
 • More precisely, the function will extract from p the polynomials of $K[{U}_{1},...,{U}_{d}]$, which are the coefficients of p when regarded as a polynomial in $\left(K[{U}_{1},...,{U}_{d}]\right)[{X}_{1},...,{X}_{n}]$. Moreover, the extracted polynomials might be simplified while preserving the variety defined by them.
 • This command is part of the RegularChains[ParametricSystemTools] package, so it can be used in the form CoefficientsInParameters(..) only after executing the command with(RegularChains[ParametricSystemTools]). However, it can always be accessed through the long form of the command by using RegularChains[ParametricSystemTools][CoefficientsInParameters](..).

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $\mathrm{with}\left(\mathrm{ParametricSystemTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[x,a,b,c\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)
 > $p≔a{x}^{2}+{a}^{2}x+bc$
 ${p}{≔}{{a}}^{{2}}{}{x}{+}{a}{}{{x}}^{{2}}{+}{b}{}{c}$ (2)
 > $\mathrm{CoefficientsInParameters}\left(p,3,R\right)$
 $\left[{a}{,}{b}{}{c}\right]$ (3)