Using Gröbner bases to determine algebra membership, split surjective algebra homomorphisms determine birational equivalence
Let \(X_ 1,...,X_ n\), \(T_ 1,...,T_ m\) be indeterminates over the field k and let \(g,f_ 1,...,f_ m\in k[X_ 1,...,X_ n]\). An algorithm involving Gröbner bases is given which will determine if \(g\in k[f_ 1,...,f_ m]\), and if so will produce \(P\in k[T_ 1,...,T_ m]\) such that \(g=P(f_ 1,...,f_ m)\). Thus for a k-algebra homomorphism \(\omega\) : \(B\to k[X_ 1,...,X_ n]\), where B is a finitely generated k-algebra, the algorithm will determine if \(\omega\) is onto, and if so it will produce a right inverse of \(\omega\). To explain the algorithm further choose a term ordering on the monomials of \(k[X_ 1,...,X_ n\), \(T_ 1,...,T_ m]\) such that each \(X_ i\) is larger than any monomial in \(k[T_ 1,...,T_ m]\). Let J be the ideal of \(k[X_ 1,...,X_ n\), \(T_ 1,...,T_ m]\) generated by \(\{f_ 1-T_ 1,...,f_ m-X_ m\}\), and let G be a Gröbner basis for J. Then for \(g\in k[X_ 1,...,X_ n]\) we may reduce g over G until reduction terminates, say with \(h\in k[X_ 1,...,X_ n\), \(T_ 1,...,T_ m]\). Then \(g\in k[X_ 1,....,X_ n]\) if and only if \(h\in k[T_ 1,...,T_ m]\) and in this case \(g=h(f_ 1,...,f_ m)\). Some related results given are tests for when \(k[f_ 1,...,f_ m]=k[X_ 1,...,X_ n]\) and when \(k(f_ 1,...,f_ m)=k(X_ 1,...,X_ n)\).
scientific article; zbMATH DE number 4115957