{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,5,18]],"date-time":"2023-05-18T06:14:20Z","timestamp":1684390460830},"reference-count":27,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2018,8,3]],"date-time":"2018-08-03T00:00:00Z","timestamp":1533254400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2019,3]]},"abstract":"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 Gr\u00f6bner 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 F\u00fcredi [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.<\/jats:p>","DOI":"10.1017\/s0963548318000342","type":"journal-article","created":{"date-parts":[[2018,8,3]],"date-time":"2018-08-03T07:23:12Z","timestamp":1533280992000},"page":"253-279","source":"Crossref","is-referenced-by-count":2,"title":["Bounding the Number of Common Zeros of Multivariate Polynomials and Their Consecutive Derivatives"],"prefix":"10.1017","volume":"28","author":[{"given":"O.","family":"GEIL","sequence":"first","affiliation":[]},{"given":"U.","family":"MART\u00cdNEZ-PE\u00d1AS","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2018,8,3]]},"reference":[{"key":"S0963548318000342_ref27","doi-asserted-by":"crossref","unstructured":"Zippel R. (1989) An explicit separation of relativised random and polynomial time and relativised deterministic polynomial time. 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