Related papers: Equivalence of polynomial conjectures in additive …
We study the interplay between recurrences for zeta related functions at integer values, `Minor Corner Lattice' Toeplitz determinants and integer composition based sums. Our investigations touch on functional identities due to Ramanujan and…
A classical result in additive combinatorics, which is a combination of Balog-Szemer\'edi-Gowers theorem and a variant of Freiman's theorem due to Ruzsa, says that if a subset $A$ of $\mathbb{F}_p^n$ contains at least $c |A|^3$ additive…
A combinatorial identity that was needed in Ahlgren and Ono's proof of a certain congruence conjecture of Frits Beukers is stated, and a pointer to its WZ proof is given.
The union-closed sets conjecture (Frankl's conjecture) says that for any finite union-closed family of finite sets, other than the family consisting only of the empty set, there exists an element that belongs to at least half of the sets in…
In this article, we verify the additivity for rank of a sum of coprime monomials and bivariate polynomials generalizing the result in (\cite{CCG}). We also show similar results hold for cactus rank.
In this article, applying the quasi-Gorenstein analogous of the Ulrich's deformation of certain Gorenstein rings we show that some homological conjectures, including the Monomial Conjecture, Big Cohen-Macaulay Algebra Conjecture as well as…
We add further notions to Lehmann's list of numerical analogues to the Kodaira dimension of pseudo-effective divisors on smooth complex projective varieties, and show new relations between them. Then we use these notions and relations to…
In this short paper we propose four conjectures in synthetic geometry that generalize Erdos-Mordell Theorem, and three conjectures in number theory that generalize Fermat Numbers.
In this paper, we consider a conjecture of Erdos and Rosenfeld and a conjecture of Ruzsa when the number is an almost square. By the same method, we consider lattice points of a circle close to the x-axis with special radii.
Let $p$ be a prime. One formulation of the Polynomial Freiman-Ruzsa conjecture over $\mathbb{F}_p$ can be stated as follows. If $\phi : \mathbb{F}_p^n \rightarrow \mathbb{F}_p^N$ is a function such that $\phi(x+y) - \phi(x) - \phi(y)$ takes…
The aim of this work is to prove a conjecture related to the Combinatorial Invariance Conjecture of Kazhdan-Lusztig polynomials, in the parabolic setting, for lower intervals in every arbitrary Coxeter group. This result improves and…
This paper studies self-injective algebras of polynomial growth. We prove that the derived equivalence classification of weakly symmetric algebras of domestic type coincides with the classification up to stable equivalences (of Morita…
Fox's conjecture (1962) states that the sequence of absolute values of the coefficients of the Alexander polynomial of alternating links is trapezoidal. While the conjecture remains open in general, a number of special cases have been…
We prove a recent conjecture by Ulas on reducible polynomial substitutions.
The log-rank conjecture is a longstanding open problem with multiple equivalent formulations in complexity theory and mathematics. In its linear-algebraic form, it asserts that the rank and partitioning number of a Boolean matrix are…
We prove an effective version of the inverse theorem for the Gowers $U^3$-norm for functions supported on high-rank quadratic level sets in finite vector spaces. For configurations controlled by the $U^3$-norm (complexity-two…
Motivated by Sato and Mori's work on the Korteweg-de Vries (KdV) equation and the modified KdV equation, Mizukawa, Nakajima, and Yamada made a conjecture on 2-reduced Schur functions and Schur's Q-functions. The conjecture claims that…
In 2023, Gullerud, Johnson, and Mbirika presented results on their study of certain tridiagonal real symmetric matrices. As part of their work, they studied the roots to nonhomogeneous equations related to characteristic polynomials of…
We derive a lower bound for a second moment of the reciprocal of the derivative of the Riemann zeta-function averaged over the zeros of the zeta-function that is half the size of the conjectured value. Our result is conditional upon the…
We establish new bounds in the Bogolyubov-Ruzsa lemma, demonstrating that if A is a subset of a finite abelian group with density alpha, then 3A-3A contains a Bohr set of rank O(log^2 (2/alpha)) and radius Omega(log^{-2} (2/alpha)). The…