Related papers: Toric Generalized Characteristic Polynomials
A polyhedral method to solve a system of polynomial equations exploits its sparse structure via the Newton polytopes of the polynomials. We propose a hybrid symbolic-numeric method to compute a Puiseux series expansion for every space curve…
We present a novel, general, and unifying point of view on sparse approaches to polynomial optimization. Solving polynomial optimization problems to global optimality is a ubiquitous challenge in many areas of science and engineering.…
A new efficient algorithm is proposed for factoring polynomials over an algebraic extension field. The extension field is defined by a polynomial ring modulo a maximal ideal. If the maximal ideal is given by its Groebner basis, no extra…
Polynomial system solving arises in many application areas to model non-linear geometric properties. In such settings, polynomial systems may come with degeneration which the end-user wants to exclude from the solution set. The…
We present a modification of Newton's method to restore quadratic convergence for isolated singular solutions of polynomial systems. Our method is symbolic-numeric: we produce a new polynomial system which has the original multiple solution…
In this paper, a new method to determine the complete set of dynamically balanced planar four-bar mechanims is presented. Using complex variables to model the kinematics of the mechanism, the dynamic balancing constraints are written as…
We present fully polynomial approximation schemes for a broad class of Holant problems with complex edge weights, which we call Holant polynomials. We transform these problems into partition functions of abstract combinatorial structures…
Generalized Fourier series with orthogonal polynomial bases have useful applications in several fields, including differential equations, pattern recognition, and image and signal processing. However, computing the generalized Fourier…
Recent advancements in quantum computing and quantum-inspired algorithms have sparked renewed interest in binary optimization. These hardware and software innovations promise to revolutionize solution times for complex problems. In this…
The question about maximal size of independent system of word equations is one of the most striking problems in combinatorics on words. Recently, Aleksi Saarela has introduced a new approach to the problem that is based on linear-algebraic…
Many statistical models are algebraic in that they are defined by polynomial constraints or by parameterizations that are polynomial or rational maps. This opens the door for tools from computational algebraic geometry. These tools can be…
We construct a convergent family of outer approximations for the problem of optimizing polynomial functions over convex bodies subject to polynomial constraints. This is achieved by generalizing the polarization hierarchy, which has…
We give a high precision polynomial-time approximation scheme for the supremum of any honest n-variate (n+2)-nomial with a constant term, allowing real exponents as well as real coefficients. Our complexity bounds count field operations and…
The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear…
In this thesis, a new class of algorithms based on Sums of Squares Programming is developed. These allow to reduce a degree-$d$ homogeneous polynomial $T = \sum_{i = 1}^m \langle a_i, X \rangle^d $ to a quadratic form being close to a…
We consider systems of polynomial equations and inequalities in $\mathbb{Q}[\boldsymbol{y}][\boldsymbol{x}]$ where $\boldsymbol{x} = (x_1, \ldots, x_n)$ and $\boldsymbol{y} = (y_1, \ldots,y_t)$. The $\boldsymbol{y}$ indeterminates are…
A fundamental problem in computer science is to find all the common zeroes of $m$ quadratic polynomials in $n$ unknowns over $\mathbb{F}_2$. The cryptanalysis of several modern ciphers reduces to this problem. Up to now, the best complexity…
We improve and simplify the result of the part 4 of "Counting curves and their projections" (Joachim von zur Gathen, Marek Karpinski, Igor Shparlinski) by showing that counting roots of a sparse polynomial over $\mathbb{F}_{2^n}$ is #P- and…
In this paper methods for simultaneous finding all roots of generalized polynomials are developed. These methods are related to the case when the roots are multiple. They possess cubic rate of convergence and they are as labour-consuming as…
We investigate the computational complexity of problems on toric ideals such as normal forms, Gr\"obner bases, and Graver bases. We show that all these problems are strongly NP-hard in the general case. Nonetheless, we can derive efficient…