On the number of zeros of multiplicity r

Details On the number of zeros of multiplicity r On the number of zeros of multiplicity r

2009-12-08

arxiv.org/abs/0912.1436v2

Mathematics

Number Theory

21 pages, 1 figure, 9 tables. More tables available at webpage listed as reference [8] in the paper

Scientific paper

Let S be a finite subset of a field. For multivariate polynomials the generalized Schwartz-Zippel bound [2], [4] estimates the number of zeros over Sx...xS counted with multiplicity. It does this in terms of the total degree, the number of variables and |S|. In the present work we take into account what is the leading monomial. This allows us to consider more general point ensembles and most importantly it allows us to produce much more detailed information about the number of zeros of multiplicity r than can be deduced from the generalized Schwartz-Zippel bound. We present both upper and lower bounds.

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