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Using Gröbner bases to determine algebra membership, split surjective algebra homomorphisms determine birational equivalence. (English) Zbl 0681.68052
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 $$k(f_ 1,...,f_ m)=k(X_ 1,...,X_ n)$$.
Reviewer: D.E.Rush

MSC:
 68W30 Symbolic computation and algebraic computation 13B25 Polynomials over commutative rings
Gröbner bases
Full Text:
References:
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