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.

ON THE DISTRIBUTION OF POINTS ON THE GENERALIZED MARKOFF–HURWITZ AND DWORK HYPERSURFACES

2014 ◽  
Vol 10 (01) ◽  
pp. 151-160 ◽  
Author(s):  
IGOR E. SHPARLINSKI
Keyword(s):  
Exponential Sums ◽  
Side Length ◽  
Character Sum ◽  
Distribution Of Points

We use bounds of mixed character sum modulo a prime p to study the distribution of points on the hypersurface [Formula: see text] for some polynomials fi ∈ ℤ[X] that are not constant modulo a prime p and integers ki with gcd (ki, p-1) = 1, i = 1, …, n. In the case of [Formula: see text] the above congruence is known as the Markoff–Hurwitz hypersurface, while for [Formula: see text] it is known as the Dwork hypersurface. In particular, we obtain non-trivial results about the number of solution in boxes with the side length below p1/2, which seems to be the limit of more general methods based on the bounds of exponential sums along varieties.

On a variant of sum-product estimates and explicit exponential sum bounds in prime fields

2009 ◽  
Vol 146 (1) ◽  
pp. 1-21 ◽  
Author(s):  
J. BOURGAIN ◽  
M. Z. GARAEV
Keyword(s):  
Prime Order ◽  
Exponential Sums ◽  
Exponential Sum ◽  
Complex Numbers ◽  
Number Of Solutions ◽  
Prime Fields ◽  
Explicit Bounds ◽  
Multiplicative Subgroup ◽  
Image Position

AbstractLet Fp be the field of a prime order p and F*p be its multiplicative subgroup. In this paper we obtain a variant of sum-product estimates which in particular implies the bound for any subset A ⊂ Fp with 1 < |A| < p12/23. Then we apply our estimate to obtain explicit bounds for some exponential sums in Fp. We show that for any subsets X, Y, Z ⊂ F*p and any complex numbers αx, βy, γz with |αx| ≤ 1, |βy| ≤ 1, |γz| ≤ 1, the following bound holds: We apply this bound further to show that if H is a subgroup of F*p with |H| > p1/4, then Finally we show that if g is a generator of F*p then for any M < p the number of solutions of the equation is less than $M^{3-1/24+o(1)}\Bigl(1+(M^2/p)^{1/24}\Bigr).$. This implies that if p1/2 < M < p, then

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 SOLUTIONS TO POLYNOMIAL CONGRUENCES IN WELL-SHAPED SETS

2013 ◽  
Vol 88 (3) ◽  
pp. 435-447 ◽  
Author(s):  
BRYCE KERR
Keyword(s):  
General Class ◽  
Convex Sets ◽  
Mean Value ◽  
Acta Arith ◽  
Mean Value Theorem ◽  
Number Of Solutions ◽  
Polynomial Congruences ◽  
Irregularities Of Distribution

AbstractWe use a generalisation of Vinogradov’s mean value theorem of Parsell et al. [‘Near-optimal mean value estimates for multidimensional Weyl sums’, arXiv:1205.6331] and ideas of Schmidt [‘Irregularities of distribution. IX’, Acta Arith. 27 (1975), 385–396] to give nontrivial bounds for the number of solutions to polynomial congruences, when the solutions lie in a very general class of sets, including all convex sets.

Synthesis of Watt-type Timed Curve Generators and Selection from Continuous Cognate Spaces

2021 ◽  
pp. 1-11
Author(s):  
Aravind Baskar ◽  
Mark Plecnik
Keyword(s):  
Recent Work ◽  
Polynomial Systems ◽  
Number Of Solutions ◽  
End Effector ◽  
Continuous Space ◽  
Trade Offs ◽  
Synthesis Of Mechanisms ◽  
Free Parameters ◽  
Selection Of

Abstract Following recent work on Stephenson-type mechanisms, the synthesis equations of Watt six-bar mechanisms that act as timed curve generators are formulated and systematically solved. Four variations of the problem arise by assigning the actuator and end effector onto different links. The approach produces exact synthesis of mechanisms up to eight precision points. Polynomial systems are formulated and their maximum number of solutions is estimated using the algorithm of random monodromy loops. Certain variations of Watt timed curve generators possess free parameters that do not affect the output motion, indicating a continuous space of cognate mechanisms. Packaging compactness, clearance, and dimensional sensitivity are characterized across the cognate space to illustrate trade-offs and aid in selection of a final mechanism.

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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