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](..).

$\mathrm{with}\left(\mathrm{RegularChains}\right)\:$

$\mathrm{with}\left(\mathrm{ParametricSystemTools}\right)\:$

$R≔\mathrm{PolynomialRing}\left(\left[x\,a\,b\,c\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$p≔a{x}^2+a^{2}x+bc$

$p≔a^{2}x+a{x}^2+bc$

$\mathrm{CoefficientsInParameters}\left(p\,3\,R\right)$

$\left[a\,bc\right]$