相关论文: Solving binomial Thue equations
We consider the equation $P(Q(x_1,\ldots,x_\nu))=Q(P(x_1),\ldots,P(x_\nu))$ in polynomials over the field of complex numbers and prove that if ${\rm deg}(P)>1$, then it is only solvable in polynomials that are affinely conjugate to…
We present a new algorithm for computing hyperexponential solutions of ordinary linear differential equations with polynomial coefficients. The algorithm relies on interpreting formal series solutions at the singular points as analytic…
In this semi-expository article we solve the diophantine equation $m^5+(4 \cdot 5^4 b^4)mn^4 - n^5=1$ for all integers $b \neq 0$. This gives an example of a family of quintic Thue equations that can be solved completely by using nothing…
We consider discrete best approximation problems in the setting of tropical algebra, which is concerned with the theory and application of algebraic systems with idempotent operations. Given a set of input--output pairs of an unknown…
We are concerned with the tensor equations whose coefficient tensor is an M-tensor. We first propose a Newton method for solving the equation with a positive constant term and establish its global and quadratic convergence. Then we extend…
Two-step predictor/corrector methods are provided to solve three classes of problems that present themselves as systems of ordinary differential equations (ODEs). In the first class, velocities are given from which displacements are to be…
We show that the binomial and related multiplicative character sums $$ \sum_{\stackrel{x=1}{(x,p)=1}}^{p^m} \chi (x^l(Ax^k +B)^w),\hspace{3ex} \sum_{x=1}^{p^m} \chi_1 (x)\chi_2(Ax^k +B), $$ have a simple evaluation for large enough $m$ (for…
Let $a,b,c$ be fixed coprime positive integers with $\min\{a,b,c\}>1$. In this paper, combining the Gel'fond-Baker method with an elementary approach, we prove that if $\max\{a,b,c\}>5\times 10^{27}$, then the equation $a^x+b^y=c^z$ has at…
We focus on computing certified upper bounds for the positive maximal singular value (PMSV) of a given matrix. The PMSV problem boils down to maximizing a quadratic polynomial on the intersection of the unit sphere and the nonnegative…
The most common procedure to solve a linear bilevel problem in the PES community is, by far, to transform it into an equivalent single-level problem by replacing the lower level with its KKT optimality conditions. Then, the complementarity…
We establish upper bounds of bit complexity of computing solution operators for symmetric hyperbolic systems of PDEs. Here we continue the research started in in our revious publications where computability, in the rigorous sense of…
This paper deals with the use of numerical methods based on random root sampling techniques to solve some theoretical problems arising in the analysis of polynomials. These methods are proved to be practical and give solutions where…
Optimization over $l\times m\times n$ integer $3$-way tables with given line-sums is NP-hard already for fixed $l=3$, but is polynomial time solvable with both $l,m$ fixed. In the {\em huge} version of the problem, the variable dimension…
We consider numerical certification of approximate solutions to a system of polynomial equations with more equations than unknowns by first certifying solutions to a square subsystem. We give several approaches that certifiably select which…
For the $n$th cyclotomic polynomial $\Phi_n$, let $A(n)$ denote the greatest absolute value of its coefficients, its height, and let $D(n)$ denote the difference between its largest and smallest coefficients, its diameter. We show that for…
Consider real or complex polynomial Riccati differential equations $a(x) \dot y=b_0(x)+b_1(x)y+b_2(x)y^2$ with all the involved functions being polynomials of degree at most $\eta$. We prove that the maximum number of polynomial solutions…
A solution for Smale's 17th problem, for the case of systems with bounded degree was recently given. This solution, an algorithm computing approximate zeros of complex polynomial systems in average polynomial time, assumed infinite…
We consider the problem of finding optimally stable polynomial approximations to the exponential for application to one-step integration of initial value ordinary and partial differential equations. The objective is to find the largest…
In the space of all entire functions it is solved the problem of interpolation taking into account multiplicities by sums of the series of exponentials with the exponents from a given set. It is found a criterion of solubility of the…
Maximum likelihood estimation in statistics leads to the problem of maximizing a product of powers of polynomials. We study the algebraic degree of the critical equations of this optimization problem. This degree is related to the number of…