TY - JOUR
T1 - Bounding the number of common zeros of multivariate polynomials and their consecutive derivatives
AU - Geil, Hans Olav
AU - Martinez Peñas, Umberto
PY - 2019/3/1
Y1 - 2019/3/1
N2 - We upper-bound the number of common zeros over a finite grid of multivariate polynomials and an arbitrary finite collection of their consecutive Hasse derivatives (in a coordinate-wise sense). To that end, we make use of the tool from Gröbner basis theory known as footprint. Then we establish and prove extensions in this context of a family of well-known results in algebra and combinatorics. These include Alon's combinatorial Nullstellensatz [1], existence and uniqueness of Hermite interpolating polynomials over a grid, estimations of the parameters of evaluation codes with consecutive derivatives [20], and bounds on the number of zeros of a polynomial by DeMillo and Lipton [8], Schwartz [25], Zippel [26, 27] and Alon and Füredi [2]. As an alternative, we also extend the Schwartz-Zippel bound to weighted multiplicities and discuss its connection to our extension of the footprint bound.
AB - We upper-bound the number of common zeros over a finite grid of multivariate polynomials and an arbitrary finite collection of their consecutive Hasse derivatives (in a coordinate-wise sense). To that end, we make use of the tool from Gröbner basis theory known as footprint. Then we establish and prove extensions in this context of a family of well-known results in algebra and combinatorics. These include Alon's combinatorial Nullstellensatz [1], existence and uniqueness of Hermite interpolating polynomials over a grid, estimations of the parameters of evaluation codes with consecutive derivatives [20], and bounds on the number of zeros of a polynomial by DeMillo and Lipton [8], Schwartz [25], Zippel [26, 27] and Alon and Füredi [2]. As an alternative, we also extend the Schwartz-Zippel bound to weighted multiplicities and discuss its connection to our extension of the footprint bound.
UR - http://www.scopus.com/inward/record.url?scp=85052625888&partnerID=8YFLogxK
U2 - 10.1017/S0963548318000342
DO - 10.1017/S0963548318000342
M3 - Journal article
SN - 0963-5483
VL - 28
SP - 253
EP - 279
JO - Combinatorics, Probability & Computing
JF - Combinatorics, Probability & Computing
IS - 2
ER -