return the coefficients of a polynomial with respect to parameters
CoefficientsInParameters(p, d, R)
polynomial in the ring
number of parameters
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, U1,...,Ud, and the other variables, X1,...,Xn, are regarded as unknowns.
The common zeros of the polynomials in lp form the variety of Kd where the polynomial p is identically zero when regarded as a polynomial in X1,...,Xn with coefficients in K[U1,...,Ud] where K is the algebraic closure of the ground field of R.
More precisely, the function will extract from p the polynomials of K[U1,...,Ud], which are the coefficients of p when regarded as a polynomial in K[U1,...,Ud][X1,...,Xn]. 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](..).
R ≔ PolynomialRing⁡x,a,b,c
R ≔ polynomial_ring
p ≔ a⁢x2+a2⁢x+b⁢c
p ≔ a2⁢x+a⁢x2+b⁢c
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