# QEtaGroebner(C, E)¶

QEtaGroebner(C, E) is a wrapper around the GrobnerPackage. It basically converts given polynomials to elements in PolynomialRing(C, E), does some Groebner basis computation or reduction and then converts back.

extendedNormalForm: (Polynomial C, List Polynomial C, List Symbol, Symbol, List Symbol) -> Polynomial C

extendedNormalForm(pol, gb, syms, f, gsyms) reduces pol with respect to gb and returns that reduced polynomial together with its relations in terms of gb. The original polynomial is represented by the variable f and the Groebner basis elements by the variables gsyms. We assume that each of the polynomials pol and gb is indeed a polynomial in syms (and no other variables) and that E corresponds to exactly #syms variables.

extendedNormalForm: (Polynomial C, List Polynomial C, List Symbol, Symbol, String) -> Polynomial C

extendedNormalForm(pol, gb, syms, f, g) returns extendedNormalForm(pol, gb, syms, f, gsyms) for gsyms := indexedSymbols(g, \#gb).

extendedNormalForms: (List Polynomial C, List Polynomial C, List Symbol, String, String) -> List Polynomial C

extendedNormalForms(lpol, gb, syms, f, g) returns [extendedNormalForm(p, gb, syms, f, g) for p in lpol for f in indexedSymbols(f, #lpol)]

groebner: (List Polynomial C, List Symbol) -> List Polynomial C

groebner(lpol, syms) considers the list of polynomials as polynomials in syms, and computes a Groebner basis with respect to the order given by E. We assume that each of the polynomials in lpol is indeed a polynomial in syms (and no other variables) and that E corresponds to exactly #syms variables.

groebnerExtend: (List Polynomial C, List Polynomial C, List Symbol) -> List Polynomial C

groebnerExtend(lpol, gb, syms) computes a Groebner basis of the union of lpos and gb under the assumption that gb is already a Groebner basis wrt. the order given by E. We assume that each of the polynomials in lpol and gb is indeed a polynomial in syms (and no other variables) and that E corresponds to exactly #syms variables.

normalForms: (List Polynomial C, List Polynomial C, List Symbol) -> List Polynomial C

normalForms(lpol, gb, syms) reduces each polynomial from lpol with respect to the Groebner basis given by gb. We assume that each of the polynomials in lpol and gb is indeed a polynomial in syms (and no other variables) and that E corresponds to exactly #syms variables.