preprint · arXiv (Cornell University) · 2016

Intersection multiplicity, Milnor number and Bernstein's theorem

Cited by 2Open access

Abstract

We explicitly characterize when the Milnor number at the origin of a polynomial or power series (over an algebraically closed field k of arbitrary characteristic) is the minimum of all polynomials with the same Newton diagram, which completes works of Kushnirenko (Invent. Math., 1976) and Wall (J. Reine Angew. Math., 1999). Given a fixed collection of n convex integral polytopes in R^n, we also give an explicit characterization of systems of n polynomials supported at these polytopes which have the maximum number (counted with multiplicity) of isolated zeroes on k^n, or more generally, on a union of torus orbits on k^n; this completes the program (undertaken by many authors including Khovanskii (Funkcional. Anal. i Prilozen, 1978), Huber and Sturmfels (Discrete Comput. Geom., 1997), Rojas (J. Pure Appl. Algebra, 1999)) of the extension to k^n of Bernstein's theorem (Funkcional. Anal. i Prilozen, 1975) on number of solutions of n polynomials on (k^*)^n. Our solutions to these two problems are connected by the computation of the intersection multiplicity at the origin of n hypersurfaces determined by n generic polynomials.

Cite this paper

APA 7

Mondal, P. (2016). Intersection multiplicity, Milnor number and Bernstein's theorem. arXiv (Cornell University). https://doi.org/10.48550/arxiv.1607.04860

MLA 9

Mondal, Pinaki. “Intersection multiplicity, Milnor number and Bernstein's theorem.” arXiv (Cornell University), 2016, https://doi.org/10.48550/arxiv.1607.04860.

Chicago

Mondal, Pinaki. 2016. “Intersection multiplicity, Milnor number and Bernstein's theorem.” arXiv (Cornell University). https://doi.org/10.48550/arxiv.1607.04860.

Harvard

Mondal, P. (2016) ‘Intersection multiplicity, Milnor number and Bernstein's theorem’, arXiv (Cornell University). Available at: https://doi.org/10.48550/arxiv.1607.04860.

BibTeX

@misc{mondal2016intersection,
  title = {{Intersection multiplicity, Milnor number and Bernstein's theorem}},
  author = {Mondal, Pinaki},
  journal = {arXiv (Cornell University)},
  year = {2016},
  doi = {10.48550/arxiv.1607.04860}
}

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