On the Number of Solutions to Polynomial Systems of Equations

1980 ◽  
Vol 17 (4) ◽  
pp. 540-546 ◽  
Author(s):  
C. B. Garcia ◽  
T. Y. Li
Keyword(s):  
Polynomial Systems ◽  
Systems Of Equations ◽  
Number Of Solutions

scholarly journals Accelerated Solution of Multivariate Polynomial Systems of Equations

2003 ◽  
Vol 32 (2) ◽  
pp. 435-454 ◽  
Author(s):  
B. Mourrain ◽  
V. Y. Pan ◽  
O. Ruatta
Keyword(s):  
Polynomial Systems ◽  
Systems Of Equations ◽  
Multivariate Polynomial

scholarly journals ON SOLUTIONS TO SOME POLYNOMIAL CONGRUENCES IN SMALL BOXES

2013 ◽  
Vol 89 (2) ◽  
pp. 300-307
Author(s):  
IGOR E. SHPARLINSKI
Keyword(s):  
Exponential Sums ◽  
Polynomial Systems ◽  
Side Length ◽  
Character Sum ◽  
Number Of Solutions ◽  
Polynomial Congruences

AbstractWe use bounds of mixed character sum to study the distribution of solutions to certain polynomial systems of congruences modulo a prime $p$. In particular, we obtain nontrivial results about the number of solutions in boxes with the side length below ${p}^{1/ 2} $, which seems to be the limit of more general methods based on the bounds of exponential sums along varieties.

Solving Polynomial Systems for the Kinematic Analysis and Synthesis of Mechanisms and Robot Manipulators

1995 ◽  
Vol 117 (B) ◽  
pp. 71-79 ◽  
Author(s):  
M. Raghavan ◽  
B. Roth
Keyword(s):  
Kinematic Analysis ◽  
State Of The Art ◽  
Polynomial Systems ◽  
The State ◽  
Polynomial Equations ◽  
Systems Of Equations ◽  
Synthesis Of Mechanisms ◽  
Analysis And Synthesis ◽  
Systems Of Polynomial Equations ◽  
And Robotics

Problems in mechanisms analysis and synthesis and robotics lead naturally to systems of polynomial equations. This paper reviews the state of the art in the solution of such systems of equations. Three well-known methods for solving systems of polynomial equations, viz., Dialytic Elimination, Polynomial Continuation, and Grobner bases are reviewed. The methods are illustrated by means of simple examples. We also review important kinematic analysis and synthesis problems and their solutions using these mathematical procedures.

On the number of solutions of multivariable polynomial systems

1983 ◽  
Vol 28 (2) ◽  
pp. 224-227 ◽  
Author(s):  
A. Benallou ◽  
D. Mellichamp ◽  
D. Seborg
Keyword(s):  
Polynomial Systems ◽  
Number Of Solutions ◽  
Multivariable Polynomial

scholarly journals How many tuples of group elements have a given property? With an appendix by Dmitrii V. Trushin

2014 ◽  
Vol 24 (04) ◽  
pp. 413-428 ◽  
Author(s):  
Anton A. Klyachko ◽  
Anna A. Mkrtchyan
Keyword(s):  
Systems Of Equations ◽  
Number Of Solutions ◽  
First Order ◽  
Group Language ◽  
The Matrix

Generalizing Solomon's theorem, Gordon and Rodriguez-Villegas have proven recently that, in any group, the number of solutions to a system of coefficient-free equations is divisible by the order of this group whenever the rank of the matrix composed of the exponent sums of ith unknown in jth equation is less than the number of unknowns. We generalize this result in two directions: first, we consider equations with coefficients, and second, we consider not only systems of equations but also any first-order formulae in the group language (with constants). Our theorem implies some amusing facts; for example, the number of group elements whose squares lie in a given subgroup is divisible by the order of this subgroup.

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