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Fixed divisor of a multivariate polynomial and generalized factorials in several variables. (English) Zbl 1405.13039

In [J. Algebra 372, 134–148 (2012; Zbl 1271.13040)] S. Evrard followed a suggestion of M. Bhargava [Am. Math. Mon. 107, No. 9, 783–799 (2000; Zbl 0987.05003)] and introduced \(v\)-orderings and generalized factorials in the case of several variables, applying it to the study of fixed divisors of multivariate polynomials. She considered a Dedekind domain \(R\) with field of fractions \(K\) and for \(S\subset R^k\) defined the generalized factorial \(k!_S\) by \[ k!_S = \{a\in R:\;a\text{Int}_k(S,R)\subset R[X], \] where \(\text{Int}_k(S,R)\) is the set of all polynomials \(f\in K[X]\) of total degree \(\leq k\) satisfying \(f(S)\subset R\). In the reviewed paper the authors present a variant of this definition taking in account the partial degrees of the considered polynomials. They prove that their generalized factorial has properties similar to Evrard’s factorial, and show on examples that their approach gives in some case better bounds for the fixed divisor of a multivariate polynomial.

MSC:

13F20 Polynomial rings and ideals; rings of integer-valued polynomials
05A10 Factorials, binomial coefficients, combinatorial functions
11B65 Binomial coefficients; factorials; \(q\)-identities
11C08 Polynomials in number theory
11R09 Polynomials (irreducibility, etc.)
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
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References:

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