Related papers: On a problem of Frobenius in three numbers

In the ring R=K[X,Y,Z]/(X^3+Y^3+Z^3), where K is a field of prime characteristic p other than 3, determining the tight closure of the ideal (X^2, Y^2, Z^2)R had existed as a classic example of the difficulty involved in tight closure…

We give an algorithm to decide whether a given braid with four strings is a product of three factors which are conjugates of standard generators of the braid group. The algorithm is of polynomial time. It is based on the Garside theory. We…

We show a new algorithm and its implementation for multiplying bit-polynomials of large degrees. The algorithm is based on evaluating polynomials at a specific set comprising a natural set for evaluation with additive FFT and a high order…

Let $a(n)$ defined by $\sum_{n=1}^{\infty}a(n)q^n := \prod_{n=1}^{\infty}\frac{1}{(1-q^{3n})(1-q^n)^3}.$ In this note, we prove that for every non-negative integer $n$, a(15n+6) \equiv 0\pmod{5}, a(15n+12) \equiv 0\pmod{5}. As a corollary,…

The theory of Frobenius groups with Frobenius complements of even order largely reduces to tractable algebraic number theory. If we consider only Frobenius complements with an upper bound $s$ on the number of distinct primes dividing the…

We present a simple randomized algorithm for approximate matrix multiplication (AMM) whose error scales with the *output* norm $\|AB\|_F$. Given any $n\times n$ matrices $A,B$ and a runtime parameter $r\leq n$, the algorithm produces in…

The abc conjecture, one of the most famous open problems in number theory, claims that three positive integers satisfying a+b=c cannot simultaneously have significant repetition among their prime factors; in particular, the product of the…

For $n\ge 3$ let $f(n)$ be the least positive integer $k$ such that $\binom nk>\frac{2^n}{n+1}$. In this paper we investigate the properties of $f(n)$.

Let $A$ be an abelian variety over a finite field $k$ with $|k|=q=p^m$. Let $\pi\in \text{End}_k(A)$ denote the Frobenius and let $v=\frac{q}{\pi}$ denote Verschiebung. Suppose the Weil $q$-polynomial of $A$ is irreducible. When…

By first solving the equation $x^3+y^3+z^3=k$ with fixed $k$ for $z$ and then considering the distance to the nearest integer function of the result, we turn the sum of three cubes problem into an optimisation one. We then apply three…

We combine the known methods for univariate polynomial root-finding and for computations in the Frobenius matrix algebra with our novel techniques to advance numerical solution of a univariate polynomial equation, and in particular…

Let $1<c<d$ be two relatively prime integers, $g_{c,d}=cd-c-d$ and $\mathbb{P}$ is the set of primes. For any given integer $k \geq 1$, we prove that $$\#\left\{p^k\le g_{c,d}:p\in \mathbb{P}, ~p^k=cx+dy,~x,y\in \mathbb{Z}_{\geqslant0}…

Let $\epsilon\in \{-1,1\}$. A sequence of prime numbers $p_1, p_2, p_3, ...$, such that $p_i=2p_{i-1}+\epsilon$ for all $i$, is called a {\it Cunningham chain} of the first or second kind, depending on whether $\epsilon =1$ or -1…

We study the zero distribution of the sum of the first $n$ polynomials satisfying a three-term recurrence whose coefficients are linear polynomials. We also extend this sum to a linear combination, whose coefficients are powers of $az+b$…

Fixed two positive integers m and e, some algorithms for computing the minimal Frobenius number and minimal genus of the set of numerical semigroups with multiplicity m and embedding dimension e are provided. Besides, the semigroups where…

We study the problems of testing isomorphism of polynomials, algebras, and multilinear forms. Our first main results are average-case algorithms for these problems. For example, we develop an algorithm that takes two cubic forms $f, g\in…

We study the Frobenius problem: given relatively prime positive integers a_1,...,a_d, find the largest value of t (the Frobenius number g(a_1,...,a_d)) such that m_1 a_1 + ... m_d a_d = t has no solution in nonnegative integers m_1,...,m_d.…

The sufficient conditions for solvability of a linear Diophantine equation $\sum_{i=1}^{n}a_ix_i=b$ (with $a_1,a_2,...,a_n\in \mathbb{N}$) in non-negative integers $x_1,x_2,...,x_n$ are given. The explicit formulas are given for Frobenius…

An affine monoid is an additive monoid which is cancellative, pointed and finitely generated. An affine monoid $\Lambda$ has the partial order defined by $\lambda \le \lambda + \mu$. The Frobenius complex is the order complex of an open…

We survey results produced from the interaction between methods in prime characteristic and combinatorial commutative algebra. We showcase results for edge ideals, toric varieties, Stanley-Reisner rings, and initial ideals that were proven…