Newton polytopes and the Bezout theorem

We study rational cuspidal curves in projective surfaces. We specify two criteria obstructing possible configurations of singular points that may occur on such curves. One criterion generalizes the result of Fernandez de Bobadilla, Luengo, Melle–Hernandez and Némethi and is based on the Bézout theorem. The other one is a generalization of the result obtained by Livingston and the author and relies on Ozsváth–Szabó inequalities for [Formula: see text]-invariants in Heegaard Floer homology. We show by means of explicit calculations that the two approaches give very similar obstructions.

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Hilbert polynomials and the Bezout Theorem

Introduction to Algebraic Geometry ◽

10.1017/cbo9780511755224.013 ◽

2010 ◽

pp. 207-234

Author(s):

Brendan Hassett

Keyword(s):

Hilbert Polynomials ◽

Bezout Theorem

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Newton polytopes of invariants of additive group actions

Journal of Pure and Applied Algebra ◽

10.1016/s0022-4049(99)00151-6 ◽

2001 ◽

Vol 156 (2-3) ◽

pp. 187-197 ◽

Cited By ~ 6

Author(s):

Harm Derksen ◽

Ofer Hadas ◽

Leonid Makar-Limanov

Keyword(s):

Group Actions ◽

Additive Group ◽

Newton Polytopes ◽

Additive Group Actions

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Pruning Algorithms for Pretropisms of Newton Polytopes

Computer Algebra in Scientific Computing - Lecture Notes in Computer Science ◽

10.1007/978-3-319-45641-6_31 ◽

2016 ◽

pp. 489-503 ◽

Cited By ~ 2

Author(s):

Jeff Sommars ◽

Jan Verschelde

Keyword(s):

Newton Polytopes ◽

Pruning Algorithms

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Systems of polynomials with at least one positive real zero

Journal of Algebra and Its Applications ◽

10.1142/s0219498820501832 ◽

2019 ◽

Vol 19 (10) ◽

pp. 2050183 ◽

Cited By ~ 1

Author(s):

Jie Wang

Keyword(s):

Polynomial Optimization ◽

Real Zero ◽

Sufficient Conditions ◽

Combinatorial Structure ◽

Multivariate Polynomials ◽

Positive Real ◽

Newton Polytopes

In this paper, we prove several theorems on systems of polynomials with at least one positive real zero based on the theory of conceive polynomials. These theorems provide sufficient conditions for systems of multivariate polynomials admitting at least one positive real zero in terms of their Newton polytopes and combinatorial structure. Moreover, a class of polynomials attaining their global minimums in the first quadrant are given, which is useful in polynomial optimization.

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