Related papers: The sharp bound for the number of real solutions t…
We obtain a good upper bound on the number of solutions of a diophantine equation arising from a strictly convex sequences of real numbers.
Given a polynomial system f associated with a simple multiple zero x of multiplicity {\mu}, we give a computable lower bound on the minimal distance between the simple multiple zero x and other zeros of f. If x is only given with limited…
We describe a new incomplete but terminating method for real root finding for large multivariate polynomials. We take an abstract view of the polynomial as the set of exponent vectors associated with sign information on the coefficients.…
Consider the nonlinear matrix equation X-sum_{i=1}^{m}A_{i}^{*}X^{-1}A_{i}=Q. This paper shows that there exists a unique positive definite solution to the equation without any restriction on A_{i}. Three perturbation bounds for the unique…
It is well known that the most challenging question in optimization and discrete geometry is whether there is a strongly polynomial time simplex algorithm for linear programs (LPs). This paper gives a positive answer to this question by…
An inhomogeneous Tricomi equation is considered in a strip with a polynomial right-hand side. It is shown that the Dirichlet boundary value problem with polynomial boundary conditions has a polynomial solution. An algorithm for constructing…
We show that positivity on $\mathbb{R}_+^n$ and on $\mathbb{R}^n$ of real symmetric polynomials of degree at most $p$ in $n\ge2$ variables is solvable by algorithms running in $\mathrm{poly}(n)$ time. For real symmetric quartics, we find…
We prove, for a sufficiently small subset $\mathcal{A}$ of a prime residue field, an estimate on the number of solutions to the equation $(a_1-a_2)(a_3-a_4) = (a_5-a_6)(a_7-a_8)$ with all variables in $\mathcal{A}$. We then derive new…
We study systems of $n \geq 1$ discrete differential equations of order $k\geq1$ in one catalytic variable and provide a constructive and elementary proof of algebraicity of their solutions. This yields effective bounds and a systematic…
We establish sharp estimates that adapt the polynomial method to arbitrary varieties. These include a partitioning theorem, estimates on polynomials vanishing on fixed sets and bounds for the number of connected components of real algebraic…
Many problems give rise to polynomial systems. These systems often have several parameters and we are interested to study how the solutions vary when we change the values for the parameters. Using predictor-corrector methods we track the…
Complex polynomial optimization has recently gained more and more attention in both theory and practice. In this paper, we study the optimization of a real-valued general conjugate complex form over various popular constraint sets including…
Mathematical modeling is a standard approach to solve many real-world problems and {\em diversity} of solutions is an important issue, emerging in applying solutions obtained from mathematical models to real-world problems. Many studies…
We present a new, far simpler family of counter-examples to Kushnirenko's Conjecture. Along the way, we illustrate a computer-assisted approach to finding sparse polynomial systems with maximally many real roots, thus shedding light on the…
This survey paper was primarily written as as the support for a course pesented at the JNCF2025: it aims to present some material that illustrates the kind of estimates one can obtain in effective algebraic geometry, for affine polynomial…
Regular chains and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. This paper proposes adaptations of these tools focusing on solutions of the real analogue:…
The multi-homogeneous Bezout number is a bound for the number of solutions of a system of multi-homogeneous polynomial equations, in a suitable product of projective spaces. Given an arbitrary, not necessarily multi-homogeneous system, one…
This paper depicts algorithms for solving the decision Boolean Satisfiability Problem. An extreme problem is formulated to analyze the complexity of algorithms and the complexity for solving it. A novel and easy reformulation as a lottery…
We consider a system of $R$ cubic forms in $n$ variables, with integer coefficients, which define a smooth complete intersection in projective space. Provided $n\geq 25R$, we prove an asymptotic formula for the number of integer points in…
Numerous problems of analysis (real and complex) and geometry (analytic, algebraic, Diophantine e.a.) can be reduced to calculation of the ``number of solutions'' of systems of equations, defined by algebraic equalities and differential…