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Register a new userOn a conjecture related to integer-valued polynomials
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Victor J. W. Guo
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Using the following $_4F_3$ transformation formula $$ sum_{k=0}^{n}{-x-1choose k}^2{xchoose n-k}^2=sum_{k=0}^{n}{n+kchoose 2k}{2kchoose k}^2{x+kchoose 2k}, $$ which can be proved by Zeilbergers algorithm, we confirm some special cases of a recent conjecture of Z.-W. Sun on integer-valued polynomials.
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Integer partitions express the different ways that a positive integer may be written as a sum of other positive integers. Here we explore the analytic properties of a polynomial $f_lambda(x)$ that we call the partition polynomial for the partition $lambda$, with the hope of learning new properties of partitions. We prove a recursive formula for the derivatives of $f_lambda(x)$ involving Stirling numbers of the second kind, show that the set of integrals from 0 to 1 of a normalized version of $f_lambda(x)$ is dense in $[0,1/2]$, pose a few open questions, and formulate a conjecture relating the integral to the length of the partition. We also provide specific examples throughout to support our speculation that an in-depth analysis of partition polynomials could further strengthen our understanding of partitions.
Let $S subset R$ be an arbitrary subset of a unique factorization domain $R$ and $K$ be the field of fractions of $R$. The ring of integer-valued polynomials over $S$ is the set $mathrm{Int}(S,R)= { f in mathbb{K}[x]: f(a) in R forall a in S }.$ This article is an effort to study the irreducibility of integer-valued polynomials over arbitrary subsets of a unique factorization domain. We give a method to construct special kinds of sequences, which we call $d$-sequences. We then use these sequences to obtain a criteria for the irreducibility of the polynomials in $mathrm{Int}(S,R).$ In some special cases, we explicitly construct these sequences and use these sequences to check the irreducibility of some polynomials in $mathrm{Int}(S,R).$ At the end, we suggest a generalization of our results to an arbitrary subset of a Dedekind domain.
In this paper, we address various aspects of divisibility by irreducibles in rings consisting of integer-valued polynomials. An integral domain is called atomic if every nonzero nonunit factors into irreducibles. Atomic domains that do not satisfy the ascending chain condition on principal ideals (ACCP) have proved to be elusive, and not many of them have been found since the first one was constructed by A. Grams in 1974. Here we exhibit the first class of atomic rings of integer-valued polynomials without the ACCP. An integral domain is called a finite factorization domain (FFD) if it is simultaneously atomic and an idf-domain (i.e., every nonzero element is divisible by only finitely many irreducibles up to associates). We prove that a ring is an FFD if and only if its ring of integer-valued polynomials is an FFD. In addition, we show that neither being atomic nor being an idf-domain transfer, in general, from an integral domain to its ring of integer-valued polynomials. In the same class of rings of integer-valued polynomials, we consider further properties that are defined in terms of divisibility by irreducibles, including being Cohen-Kaplansky and being Furstenberg.
It is a surprising fact that the proportion of integer lattice points visible from the origin is exactly $frac{6}{pi^2}$, or approximately 60 percent. Hence, approximately 40 percent of the integer lattice is hidden from the origin. Since 1971, many have studied a variety of problems involving lattice point visibility, in particular, searching for patterns in that 40 percent of the lattice comprised of invisible points. One such pattern is a square patch, an $n times n$ grid of $n^2$ invisible points, which we call a hidden forest. It is known that there exist arbitrarily large hidden forests in the integer lattice. However, the methods up to now involve the Chinese Remainder Theorem (CRT) on the rows and columns of matrices with prime number entries, and they have only been able to locate hidden forests very far from the origin. For example, using this method the closest known $4 times 4$ hidden forest is over 3 quintillion, or $3 times 10^{18}$, units away from the origin. We introduce the concept of quasiprime matrices and utilize a variety of computational and theoretical techniques to find some of the closest known hidden forests to this date. Using these new techniques, we find a $4 times 4$ hidden forest that is merely 184 million units away from the origin. We conjecture that every hidden forest can be found via the CRT-algorithm on a quasiprime matrix.
Let $E_n$ be the $n$-th Euler number and $(a)_n=a(a+1)cdots (a+n-1)$ the rising factorial. Let $p>3$ be a prime. In 2012, Sun proved the that $$ sum^{(p-1)/2}_{k=0}(-1)^k(4k+1)frac{(frac{1}{2})_k^3}{k!^3} equiv p(-1)^{(p-1)/2}+p^3E_{p-3} pmod{p^4}, $$ which is a refinement of a famous supercongruence of Van Hamme. In 2016, Chen, Xie, and He established the following result: $$ sum_{k=0}^{p-1}(-1)^k (3k+1)frac{(frac{1}{2})_k^3}{k!^3} 2^{3k} equiv p(-1)^{(p-1)/2}+p^3E_{p-3} pmod{p^4}, $$ which was originally conjectured by Sun. In this paper we give $q$-analogues of the above two supercongruences by employing the $q$-WZ method. As a conclusion, we provide a $q$-analogue of the following supercongruence of Sun: $$ sum_{k=0}^{(p-1)/2}frac{(frac{1}{2})_k^2}{k!^2} equiv (-1)^{(p-1)/2}+p^2 E_{p-3} pmod{p^3}. $$
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