Related papers: The binary method of integer decomposition
For polynomials of degree two which have no zeros, the method of accompanying variables is developed and zeros of associated vector polynomials are determined. Our flexible method uses a wide variety of possible vector-valued vector…
In this paper, we prove the finiteness of the number of integer solutions of the decomposable form inequalities. We also study the number of integer solutions of a sequence of decomposable form inequalities.
Algorithms and hardware for solving quadratic unconstrained binary optimization (QUBO) problems have made significant recent progress. This advancement has focused attention on formulating combinatorial optimization problems as quadratic…
In this paper, we study the "sum composition problem" between two lists $A$ and $B$ of positive integers. We start by saying that $B$ is "sum composition" of $A$ when there exists an ordered $m$-partition $[A_1,\ldots,A_m]$ of $A$ where $m$…
We present a detailed description of the recent idea for a direct decomposition of Feynman integrals onto a basis of master integrals by projections, as well as a direct derivation of the differential equations satisfied by the master…
We consider so-called squaring the square-puzzles where a given square (or rectangle) should be dissected into smaller squares. For a specific instance of such problems we demonstrate that a mathematically rigorous solution can be quite…
Using the circle method, we obtain asymptotic formulae for the number of integer solutions to certain quadratic polynomials that are uniform in the coefficients of the polynomial.
This paper presents a novel algorithm for constructing a sum-of-squares (SOS) decomposition for positive semi-definite polynomials with rational coefficients. Unlike previous methods that typically yield SOS decompositions with…
Using modular forms we determine formulas for the number of representations of a positive integer by diagonal octonary quadratic forms with coefficients $1$, $2$, $3$ or $6$.
Lagrangian duality in mixed integer optimization is a useful framework for problems decomposition and for producing tight lower bounds to the optimal objective, but in contrast to the convex counterpart, it is generally unable to produce…
We have proved in this paper that numbers can be expressed in algebraic form using one variable and two real rational quantities and thus sum of three cubes can also be expressed in algebraic form as a cubic polynomial. Using skeletal or…
We present a new tool to compute the number $\phi_\A (\b)$ of integer solutions to the linear system $$ \x \geq 0 \qquad \A \x = \b $$ where the coefficients of $\A$ and $\b$ are integral. $\phi_\A (\b)$ is often described as a \emph{vector…
Let $m$ be a positive integer and $b_{m}(n)$ be the number of partitions of $n$ with parts being powers of 2, where each part can take $m$ colors. We show that if $m=2^{k}-1$, then there exists the natural density of integers $n$ such that…
The subject matter of this work is quadratic and cubic polynomial functions with integer coefficients;and all of whose roots are integers. The material of this work is directed primarily at educators,students,and teachers of…
A non-iterative method is presented for the factorization step of sector decomposition method, which separates infrared divergent part from loop integration. This method is based on a classification of asymptotic behavior of polynomials.…
We analyze the point decomposition problem (PDP) in binary elliptic curves. It is known that PDP in an elliptic curve group can be reduced to solving a particular system of multivariate non-linear system of equations derived from the so…
Decoupling multivariate polynomials is useful for obtaining an insight into the workings of a nonlinear mapping, performing parameter reduction, or approximating nonlinear functions. Several different tensor-based approaches have been…
Image restoration refers to the process of reconstructing noisy, destroyed, or missing parts of an image, which is an ill-posed inverse problem. A specific regularization term and image degradation are typically assumed to achieve…
A numerical irreducible decomposition for a polynomial system provides representations for the irreducible factors of all positive dimensional solution sets of the system, separated from its isolated solutions. Homotopy continuation methods…
We study the following problem. Given a multiset $M$ of non-negative integers, decide whether there exist and, in the positive case, compute two non-trivial multisets whose Minkowski sum is equal to $M$. The Minkowski sum of two multisets A…