# Some bounds for the construction of Gröbner bases

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## Abstract

Let *R*=*K*[*X*_{1}, ..., *X*_{ n }] be a polynomial ring over a field. For any finite subset *F* of *R*, we put *m*=‖*F*‖, *d*=*max*(*deg*(*F*) : *f* ε *F*), and we let *s* be the maximal size of the coefficients of all *f* ε *F. G*=*GB*(*F*) denotes the unique reduced Gröbner basis for the ideal (*F*) (see [B3]). We show that the number *m*′=‖*G*‖ of polynomials in *G* and their maximal degree *d*^{ t } as well as the length of the computation of *G* from *F* (with unit cost operations in *K*) are bounded recursively in (*n, m, d*). The same applies to the degrees of the polynomials occuring during the computation. Moreover, for fixed (*n, m, d*), *G* can be computed from *F* in polynomial time and linear space, when the operations of *K* can be performed in polynomial time and linear space; in addition, the vector space dimension of the residue ring *R*/(*F*) is computably stable under variation of the coefficients of polynomials in F. Corresponding facts hold for polynomial rings over commutative regular rings (see [We']) and non-commutative polynomial rings of solvable type over fields (see [KRW]). Our method does not apply to polynomial rings over *Z* or other Euclidean rings; in fact, we show that over *Z*, the length of the computation of *G* from *F* with unit cost operations in *Z does* depend on *s*.

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