scholarly journals Solving parametric systems of polynomial equations over the reals through Hermite matrices

2021 ◽  
Author(s):  
Huu Phuoc Le ◽  
Mohab Safey El Din
Keyword(s):  
Polynomial Equations ◽  
Systems Of Polynomial Equations ◽  
Parametric Systems

Construction of systems of polynomial equations with exact p-Divisibility via the covering method

2014 ◽  
Vol 13 (06) ◽  
pp. 1450013 ◽  
Author(s):  
Francis N. Castro ◽  
Ivelisse M. Rubio
Keyword(s):  
Exponential Sums ◽  
Polynomial Equations ◽  
Prime Field ◽  
Elementary Method ◽  
Covering Method ◽  
Systems Of Polynomial Equations

We present an elementary method to compute the exact p-divisibility of exponential sums of systems of polynomial equations over the prime field. Our results extend results by Carlitz and provide concrete and simple conditions to construct families of polynomial equations that are solvable over the prime field.

scholarly journals Linear Network Codes and Systems of Polynomial Equations

2008 ◽  
Vol 54 (5) ◽  
pp. 2303-2316 ◽  
Author(s):  
Randall Dougherty ◽  
Chris Freiling ◽  
Kenneth Zeger
Keyword(s):  
Polynomial Equations ◽  
Linear Network ◽  
Network Codes ◽  
Systems Of Polynomial Equations

scholarly journals Bivariate systems of polynomial equations with roots of high multiplicity

2021 ◽  
Author(s):  
I. Nikitin
Keyword(s):  
High Multiplicity ◽  
Polynomial Equations ◽  
Support Sets ◽  
Multiplicity Formula ◽  
System Of Polynomial Equations ◽  
Systems Of Polynomial Equations

Given a bivariate system of polynomial equations with fixed support sets [Formula: see text] it is natural to ask which multiplicities its solutions can have. We prove that there exists a system with a solution of multiplicity [Formula: see text] for all [Formula: see text] in the range [Formula: see text], where [Formula: see text] is the set of all integral vectors that shift B to a subset of [Formula: see text]. As an application, we classify all pairs [Formula: see text] such that the system supported at [Formula: see text] does not have a solution of multiplicity higher than [Formula: see text].

scholarly journals Galois theory for general systems of polynomial equations

2019 ◽  
Vol 155 (2) ◽  
pp. 229-245 ◽  
Author(s):  
A. Esterov
Keyword(s):  
Galois Theory ◽  
General System ◽  
Mixed Volume ◽  
Polynomial Equations ◽  
Systems Of Equations ◽  
Change Of Variables ◽  
General Polynomial ◽  
General Systems ◽  
Systems Of Polynomial Equations

We prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because arbitrary systems split into reduced irreducible ones upon monomial changes of variables. In particular, our result proves the multivariate version of the Abel–Ruffini theorem: the classification of general systems of equations solvable by radicals reduces to the classification of lattice polytopes of mixed volume 4 (which we prove to be finite in every dimension). We also notice that the monodromy of every general system of equations is either symmetric or imprimitive. The proof is based on a new result of independent importance regarding dual defectiveness of systems of equations: the discriminant of a reduced irreducible square system of general polynomial equations is a hypersurface unless the system is linear up to a monomial change of variables.

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