Results 41 to 50 of about 1,707 (110)
A Duality Based Proof of the Combinatorial Nullstellensatz [PDF]
The Electronic Journal of Combinatorics, 2009 In this note we present a proof of the combinatorial nullstellensatz using simple arguments from linear algebra.
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The canonical representation of the Drinfeld curve
Mathematische Nachrichten, Volume 297, Issue 11, Page 4115-4120, November 2024.Abstract If C$C$ is a smooth projective curve over an algebraically closed field F$\mathbb {F}$ and G$G$ is a group of automorphisms of C$C$, the canonical representation of C$C$ is given by the induced F$\mathbb {F}$‐linear action of G$G$ on the vector space H0C,ΩC$H^0\left(C,\Omega _C\right)$ of holomorphic differentials on C$C$.
Lucas Laurent, Bernhard Köck
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Computation with Polynomial Equations and Inequalities arising in Combinatorial Optimization
, 2009 The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear algebra or ...
A Kehrein +19 more
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Polynomial‐exponential equations — Some new cases of solvability
Proceedings of the London Mathematical Society, Volume 129, Issue 4, October 2024.Abstract Recently, Brownawell and the second author proved a ‘non‐degenerate’ case of the (unproved) ‘Zilber Nullstellensatz’ in connexion with ‘Strong Exponential Closure’. Here, we treat some significant new cases. In particular, these settle completely the problem of solving polynomial‐exponential equations in two complex variables.
Vincenzo Mantova, David Masser
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Recognizing Graph Theoretic Properties with Polynomial Ideals [PDF]
, 2010 Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of the polynomial
De Loera, J. A. +3 more
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Alon's Nullstellensatz for multisets
, 2011 Alon's combinatorial Nullstellensatz (Theorem 1.1 from \cite{Alon1}) is one of the most powerful algebraic tools in combinatorics, with a diverse array of applications. Let $\F$ be a field, $S_1,S_2,..., S_n$ be finite nonempty subsets of $\F$.
Kós, Géza, Rónyai, Lajos
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Combinatorial Nullstellensatz approach to polynomial expansion
Acta Arithmetica, 2014 Applying techniques similar to Combinatorial Nullstellensatz we prove a lower estimate of $|f(A,B)|$ for finite subsets $A$, $B$ of a field, and polynomial $f(x,y)$ of the form $f(x,y)=g(x)+yh(x)$, where degree of $g$ is greater then degree of $h$.
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Analytic Nullstellensätze and the model theory of valued fields
Mathematische Nachrichten, Volume 297, Issue 8, Page 2873-2917, August 2024.Abstract We present a uniform framework for establishing Nullstellensätze for power series rings using quantifier elimination results for valued fields. As an application, we obtain Nullstellensätze for p$p$‐adic power series (both formal and convergent) analogous to Rückert's complex and Risler's real Nullstellensatz, as well as a p$p$‐adic analytic ...
Matthias Aschenbrenner, Ahmed Srhir
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On the existence of zero-sum subsequences of distinct lengths
, 2012 In this paper, we obtain a characterization of short normal sequences over a finite Abelian p-group, thus answering positively a conjecture of Gao for a variety of such groups.
Girard, Benjamin
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Expressing Combinatorial Problems by Systems of Polynomial Equations and Hilbert's Nullstellensatz [PDF]
Combinatorics, Probability and Computing, 2009 Systems of polynomial equations over the complex or real numbers can be used to model combinatorial problems. In this way, a combinatorial problem is feasible (e.g., a graph is 3-colourable, Hamiltonian, etc.) if and only if a related system of polynomial equations has a solution.For an infeasible polynomial system, the (complex) Hilbert ...
De Loera, J. A. +3 more
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