Related papers: Polynomial systems with few real zeroes

Polynomial convergence bounds are considered for left, right, and split preconditioned GMRES. They include the cases of Weighted and Deflated GMRES for a linear system Ax = b. In particular, the case of positive definite A is considered.…

According to the real \tau-conjecture, the number of real roots of a sum of products of sparse polynomials should be polynomially bounded in the size of such an expression. It is known that this conjecture implies a superpolynomial lower…

It has recently been shown that the problem of testing global convexity of polynomials of degree four is {strongly} NP-hard, answering an open question of N.Z. Shor. This result is minimal in the degree of the polynomial when global…

Binary polynomial optimization is equivalent to the problem of minimizing a linear function over the intersection of the multilinear set with a polyhedron. Many families of valid inequalities for the multilinear set are available in the…

The motivation of this work stems from the numerical approximation of bounded functions by polynomials satisfying the same bounds. The present contribution makes use of the recent algebraic characterization found in [B. Despr\'es, Numer.…

Though it is well known that the roots of any affine polynomial over a finite field can be computed by a system of linear equations by using a normal base of the field, such solving approach appears to be difficult to apply when the field…

We obtain a good upper bound on the number of solutions of a diophantine equation arising from a strictly convex sequences of real numbers.

The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimisation, complexity theory, and scientific computing. Motivated by recent developments in this…

We study systems of equations of the form X1 = f1(X1, ..., Xn), ..., Xn = fn(X1, ..., Xn), where each fi is a polynomial with nonnegative coefficients that add up to 1. The least nonnegative solution, say mu, of such equation systems is…

The extremal values of multivariate trigonometric polynomials are of interest in fields ranging from control theory to filter design, but finding the extremal values of such a polynomial is generally NP-Hard. In this paper, we develop…

The solving degree of a system of multivariate polynomial equations provides an upper bound for the complexity of computing the solutions of the system via Groebner bases methods. In this paper, we consider polynomial systems that are…

We prove lower bounds on the error incurred when approximating any oscillating function using piecewise polynomial spaces. The estimates are explicit in the polynomial degree and have optimal dependence on the meshwidth and frequency when…

Locating the zeros of quaternionic polynomials is a fundamental problem with significant implications across scientific and engineering disciplines, yet the noncommutative nature of quaternion multiplication makes it fundamentally more…

We exhibit an algorithm to compute the strongest polynomial (or algebraic) invariants that hold at each location of a given affine program (i.e., a program having only non-deterministic (as opposed to conditional) branching and all of whose…

We study limitations of polynomials computed by depth two circuits built over read-once polynomials (ROPs) and depth three syntactically multi-linear formulas. We prove an exponential lower bound for the size of the $\Sigma\Pi^{[N^{1/30}]}$…

Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of Gau{\ss} count the remaining ones, approximately and exactly. For polynomials in two or more…

We study the topology of the set X of the solutions of a system of two quadratic inequalities in the real projective space RP^n (e.g. X is the intersection of two real quadrics). We give explicit formulae for its Betti numbers and for those…

We study the boundary of the cone of real polynomials that can be decomposed as a sum of squares (SOS) of real polynomials. This cone is included in the cone of nonnegative polynomials and both cones share a part of their boundary, which…

In this paper we are concerned with determining lower bounds of the number of limit cycles for piecewise polynomial holomorphic systems with a straight line of discontinuity. We approach this problem with different points of view: study of…

We consider a system of the form x'=P_n(x,y)+xR_m(x,y), y'=Q_n(x,y)+yR_m(x,y), where P_n(x,y), Q_n(x,y) and R_m(x,y) are homogeneous polynomials of degrees n, n and m, respectively, with n<=m. We prove that this system has at most one limit…