A note on Diophantine systems involving three symmetric polynomials

Let $\bar{X}_{n}=(x_{1},\ldots,x_{n})$ and $\sigma_{i}(\bar{X}_{n})=\sum x_{k_{1}}\ldots x_{k_{i}}$ be $i$-th elementary symmetric polynomial. In this note we prove that there are infinitely many triples of integers $a, b, c$ such that for each $1\leq i\leq n$ the system of Diophantine equations \begin{equation*} \sigma_{i}(\bar{X}_{2n})=a, \quad \sigma_{2n-i}(\bar{X}_{2n})=b, \quad \sigma_{2n}(\bar{X}_{2n})=c \end{equation*} has infinitely many rational solutions. This result extend the recent results of Zhang and Cai, and the author. Moreover, we also consider some Diophantine systems involving sums of powers. In particular, we prove that for each $k$ there are at least $k$ $n$-tuples of integers with the same sum of $i$-th powers for $i=1,2,3$. Similar result is proved for $i=1,2,4$ and $i=-1,1,2$.Comment: to appear in J. Number Theor

kk-partial permutations and the center of the wreath product Sk≀Sn\mathcal{S}_k\wr \mathcal{S}_n algebra

We generalize the concept of partial permutations of Ivanov and Kerov and introduce $k$-partial permutations. This allows us to show that the structure coefficients of the center of the wreath product $\mathcal{S}_k\wr \mathcal{S}_n$ algebra are polynomials in $n$ with non-negative integer coefficients. We use a universal algebra $\mathcal{I}_\infty^k$ which projects on the center $Z(\mathbb{C}[\mathcal{S}_k\wr \mathcal{S}_n])$ for each $n.$ We show that $\mathcal{I}_\infty^k$ is isomorphic to the algebra of shifted symmetric functions on many alphabets

Explicit Formulae for LL-values in Positive Characteristic

We focus on the generating series for the rational special values of Pellarin's $L$-series in $1 \leq s \leq 2(q-1)$ indeterminates, and using interpolation polynomials we prove a closed form formula relating this generating series to the Carlitz exponential, the Anderson-Thakur function, and the Anderson generating functions for the Carlitz module. We draw several corollaries, including explicit formulae and recursive relations for Pellarin's $L$-series in the same range of $s$, and divisibility results on the numerators of the Bernoulli-Carlitz numbers by monic irreducibles of degrees one and two.Comment: Some helpful clarifications in the introduction. Final edit, as appears online in Math.

Permutation combinatorics of worldsheet moduli space

52 pages, 21 figures52 pages, 21 figures; minor corrections, "On the" dropped from title, matches published version52 pages, 21 figures; minor corrections, "On the" dropped from title, matches published versio

Every sufficiently large even number is the sum of two primes

The binary Goldbach conjecture asserts that every even integer greater than $4$ is the sum of two primes. In this paper, we prove that there exists an integer $K_\alpha$ such that every even integer $x > p_k^2$ can be expressed as the sum of two primes, where $p_k$ is the $k$th prime number and $k > K_\alpha$. To prove this statement, we begin by introducing a type of double sieve of Eratosthenes as follows. Given a positive even integer $x > 4$, we sift from $[1, x]$ all those elements that are congruents to $0$ modulo $p$ or congruents to $x$ modulo $p$, where $p$ is a prime less than $\sqrt{x}$. Therefore, any integer in the interval $[\sqrt{x}, x]$ that remains unsifted is a prime $q$ for which either $x-q = 1$ or $x-q$ is also a prime. Then, we introduce a new way of formulating a sieve, which we call the sequence of $k$-tuples of remainders. By means of this tool, we prove that there exists an integer $K_\alpha > 5$ such that $p_k / 2$ is a lower bound for the sifting function of this sieve, for every even number $x$ that satisfies $p_k^2 < x < p_{k+1}^2$, where $k > K_\alpha$, which implies that $x > p_k^2 \; (k > K_\alpha)$ can be expressed as the sum of two primes.Comment: 32 pages. The manuscript was edited for proper English language by one editor at American Journal Experts (Certificate Verification Key: C0C3-5251-4504-E14D-BE84). However, afterwards some changes have been made in sections 1, 6, 7 and