Related papers: Filmor Theorem for integers

Let A be a finite set of integers. For a polynomial f(x_1,...,x_n) with integer coefficients, let f(A) = {f(a_1,...,a_n) : a_1,...,a_n \in A}. In this paper it is proved that for every pair of normalized binary linear forms f(x,y)=u_1x+v_1y…

We extend Robertson's theorem to apply to frames generated by the action of a discrete, countable abelian unitary group. Within this setup we use Stone's theorem and the theory of spectral multiplicity to analyze wandering frame…

We classify the polynomials $f(x,y) \in \mathbb R[x,y]$ such that given any finite set $A \subset \mathbb R$ if $|A+A|$ is small, then $|f(A,A)|$ is large. In particular, the following bound holds : $|A+A||f(A,A)| \gtrsim |A|^{5/2}.$ The…

The binary radix expansion of a real number can be used to code the outcome of any series of coin tosses, a fact that provides an intriguing link between number theory, measure theory and statistical physics. Inspired by this fact, a…

We construct a fully faithful functor from the category of graphs to the category of fields. Using this functor, we resolve a longstanding open problem in computable model theory, by showing that for every nontrivial countable structure S,…

The Implicit and Inverse Function Theorems are special cases of a general Implicit/Inverse Function Theorem which can be easily derived from either theorem. The theorems can thus be easily deduced from each other via the generalized…

Suppose $F$ is an infinite field and let $f \in F\{X_1, \dots,X_m\}$ be a noncommutative polynomial. Partially answering a query of Makar-Limanov, we show that there are numbers $d$ and $m'$ such that, if $F$ is closed under taking $d$th…

Let $\Omega(n)$ denote the number of prime factors of $n$. We show that for any bounded $f\colon\mathbb{N}\to\mathbb{C}$ one has \[ \frac{1}{N}\sum_{n=1}^N\, f(\Omega(n)+1)=\frac{1}{N}\sum_{n=1}^N\, f(\Omega(n))+\mathrm{o}_{N\to\infty}(1).…

This paper presents an integer decomposition method. The method first writes an integer as a polynomial with 2 as variable that its coefficients are zero or one. Then, suppose that an integer is decomposed into product of such two…

In this short note, we introduce an analogue of Wilson's theorem for all nonzero elements $a_1,a_2,...,a_{q-1}$ of a finite filed $\mathbb{F}$ with $|\mathbb{F}|=q\geq 3$, as follows: $$ \sum_{1\leq i_1< i_2<...< i_k\leq…

Working in the multitape Turing model, we show how to reduce the problem of matrix transposition to the problem of integer multiplication. If transposing an $n \times n$ binary matrix requires $\Omega(n^2 \log n)$ steps on a Turing machine,…

A classical theorem of Forster asserts that a finite module $M$ of rank $\leq n$ over a Noetherian ring of Krull dimension $d$ can be generated by $n + d$ elements. We prove a generalization of this result, with "module" replaced by…

A theorem of Thompson provides a non-self-adjoint variant of the classical Schur-Horn theorem by characterizing the possible diagonal values of a matrix with given singular values. We prove an analogue of Thompson's theorem for II_1…

A deterministic algorithm for factoring $n$ using $n^{1/3+o(1)}$ bit operations is presented. The algorithm tests the divisibility of $n$ by all the integers in a short interval at once, rather than integer by integer as in trial division.…

A method of determining two factors of an odd integer without need of multiplication or division operation in iterative portion of computation is presented. It is feasible for an implementing algorithm to use only integer addition and…

Originally motivated by algebraic invariant theory, we present an algorithm to enumerate integer vectors modulo the action of a permutation group. This problem generalizes the generation of unlabeled graph up to an isomorphism. In this…

A sharp bound is obtained for the number of ways to express the monomial $X^n$ as a product of linear factors over $\mathbb{Z}/p^{\alpha}\mathbb{Z}$. The proof relies on an induction-on-scale procedure which is used to estimate the number…

The Abels-Margulis-Soifer lemma states that if a semigroup $\Gamma$ acts strongly irreducibly by linear transformations on a finite-dimensional real vector space, then any element of $\Gamma$ can be multiplied by an element of some fixed…

Farkas' Lemma is a foundational result in linear programming, with implications in duality, optimality conditions, and stochastic and bilevel programming. Its generalizations are known as theorems of the alternative. There exist theorems of…

Let \lambda be a partition of a positive integer n. Let C be a symmetric rigid tensor category over a field k of characteristic 0 or char(k)>n, and let V be an object of C. In our main result (Theorem 4.3) we introduce a finite set of…