Related papers: Zero-nonzero and real-nonreal sign determination
We present a new algorithm for solving the real roots of a bivariate polynomial system $\Sigma=\{f(x,y),g(x,y)\}$ with a finite number of solutions by using a zero-matching method. The method is based on a lower bound for bivariate…
Zeroth-order (ZO) optimization is a subset of gradient-free optimization that emerges in many signal processing and machine learning applications. It is used for solving optimization problems similarly to gradient-based methods. However, it…
Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject…
We review recent attempts at dealing with the sign problem in Monte Carlo calculations by deforming the region of integration in the path integral from real to complex fields. We discuss the theoretical foundations, the algorithmic issues…
A polynomial algorithm is obtained for the NP-complete linear ordering problem.
We develop a numerical method for solving a system of nonlinear integral equations involving two integral terms: at the current time t, one integral is taken from 0 to t, and a different integral is taken from t to infinity. We prove the…
A possible method to solve the sign problem is developed by modifying the original theory. Considering several modifications of the partition function, the observable in the original theory is reconstructed from the identity connecting the…
Three algorithms are presented that determine the existence of satisfying assignments for 3SAT Boolean satisfiability expressions. One algorithm is presented for determining an instance of a satisfying assignment, where such exists. The…
Certifying nonnegativity of polynomials is a well-known NP-hard problem with direct applications spanning non-convex optimization, control, robotics, and beyond. A sufficient condition for nonnegativity is the Sum of Squares (SOS) property,…
In this paper, we present a generic parametrization of generically zero-dimensional parametric polynomial systems. More specifically, we study the specialization properties of the Rational Univariate Representation and derive bounds on the…
In this paper, we consider the rank-one tensor completion problem. We address the question of existence and uniqueness of the rank-one solution. In particular we show that the global uniqueness over the field of real numbers can be verified…
We consider three realization problems about monic real univariate polynomials without vanishing coefficients. Such a polynomial $P:=\sum_{j=0}^db_jx^j$ defines the sign pattern $\sigma (P):=({\rm sgn}(b_d)$, $\ldots$, ${\rm sgn}(b_0))$.…
To tackle the sign problem in the simulations of systems having indefinite or complex-valued measures, we propose a new approach which yields statistical errors smaller than the crude Monte Carlo using absolute values of the original…
Binomial ideals are special polynomial ideals with many algorithmically and theoretically nice properties. We discuss the problem of deciding if a given polynomial ideal is binomial. While the methods are general, our main motivation and…
We discuss the Fermion sign problem and, by examining a very general Hubbard-Stratonovich (HS) transformation, argue that the sign problem cannot be solved with such methods. We propose a different kind of transformation which, while not…
An approach is proposed for bounding the number of zeros that solutions of linear differential systems with polynomial coefficients may have. A bound is obtained in a special case which improves upon currently existing.
NP-Complete problems have an important attribute that if one NP-Complete problem can be solved in polynomial time, all NP-Complete problems will have a polynomial solution. The 3-CNF-SAT problem is a NP-Complete problem and the primary…
We focus on rational solutions or nearly-feasible rational solutions that serve as certificates of feasibility for polynomial optimization problems. We show that, under some separability conditions, certain cubic polynomially constrained…
In this article, we show that the completion problem, i.e. the decision problem whether a partial structure can be completed to a full structure, is NP-complete for many combinatorial structures. While the gadgets for most reductions in…
In an earlier article [3], we presented an algorithm that can be used to rigorously check whether a specific cosine or sine polynomial is nonnegative in a given interval or not. The algorithm proves to be an indispensable tool in…