On the characterization of the solution set of polynomial systems via LMI techniques

2001 ◽  
Author(s):  
G. Chesi ◽  
A. Garulli
Keyword(s):  
Polynomial Systems ◽  
Solution Set

On a Congruence Related to Character Sums

1985 ◽  
Vol 28 (4) ◽  
pp. 431-439 ◽  
Author(s):  
J. H. H. Chalk
Keyword(s):  
Prime Power ◽  
Dirichlet Character ◽  
Character Sums ◽  
Arithmetic Progressions ◽  
Solution Set ◽  
Character Sum ◽  
Equal Degree

AbstractIf χ is a Dirichlet character to a prime-power modulus pα, then the problem of estimating an incomplete character sum of the form ∑1≤x≤h χ (x) by the method of D. A. Burgess leads to a consideration of congruences of the typef(x)g'(x) - f'(x)g(x) ≡ 0(pα),where fg(x) ≢ 0(p) and f, g are monic polynomials of equal degree with coefficients in Ζ. Here, a characterization of the solution-set for cubics is given in terms of explicit arithmetic progressions.

scholarly journals AE Regularity of Interval Matrices

2018 ◽  
Vol 33 ◽  
pp. 137-146
Author(s):  
Milan Hladík
Keyword(s):  
Computational Complexity ◽  
Coefficient Matrix ◽  
Solution Set ◽  
System Of Equations ◽  
Existential Quantifier ◽  
Open Problems ◽  
Linear System Of Equations ◽  
Classes Of Matrices ◽  
Interval Coefficients

Consider a linear system of equations with interval coefficients, and each interval coefficient is associated with either a universal or an existential quantifier. The AE solution set and AE solvability of the system is defined by ∀∃- quantification. The paper deals with the problem of what properties must the coefficient matrix have in order that there is guaranteed an existence of an AE solution. Based on this motivation, a concept of AE regularity is introduced, which implies that the AE solution set is nonempty and the system is AE solvable for every right-hand side. A characterization of AE regularity is discussed, and also various classes of matrices that are implicitly AE regular are investigated. Some of these classes are polynomially decidable, and therefore give an efficient way for checking AE regularity. Eventually, there are also stated open problems related to computational complexity and characterization of AE regularity.

scholarly journals The Solution Set Characterization and Error Bound for the Extended Mixed Linear Complementarity Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Hongchun Sun ◽  
Yiju Wang
Keyword(s):  
Error Bound ◽  
Linear Complementarity Problem ◽  
Complementarity Problem ◽  
Complementarity Problems ◽  
Linear Complementarity Problems ◽  
Linear Complementarity ◽  
Solution Set ◽  
Global Error Bound ◽  
Mixed Linear Complementarity Problem

For the extended mixed linear complementarity problem (EML CP), we first present the characterization of the solution set for the EMLCP. Based on this, its global error bound is also established under milder conditions. The results obtained in this paper can be taken as an extension for the classical linear complementarity problems.

scholarly journals Solving polynomial systems via homotopy continuation and monodromy

2018 ◽  
Vol 39 (3) ◽  
pp. 1421-1446 ◽  
Author(s):  
Timothy Duff ◽  
Cvetelina Hill ◽  
Anders Jensen ◽  
Kisun Lee ◽  
Anton Leykin ◽  
...  
Keyword(s):  
State Of The Art ◽  
Polynomial Systems ◽  
Solution Set ◽  
Software Implementation ◽  
Homotopy Continuation ◽  
Expected Number ◽  
Number Of Solutions ◽  
Study Methods ◽  
Software Packages ◽  
Generic System

Abstract We study methods for finding the solution set of a generic system in a family of polynomial systems with parametric coefficients. We present a framework for describing monodromy-based solvers in terms of decorated graphs. Under the theoretical . that monodromy actions are generated uniformly, we show that the expected number of homotopy paths tracked by an algorithm following this framework is linear in the number of solutions. We demonstrate that our software implementation is competitive with the existing state-of-the-art methods implemented in other software packages.

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