Related papers: Generalized Lucas congruences and linear $p$-schem…
Linear-constraint loops are programs whose transition relation is specified by a system of linear inequalities. The termination problem asks, given a loop, whether it admits an infinite computation. Decidability of termination remains open…
We answer positively a question of Church, Miller, Nagpal and Reinhold on existence of a linear bound on the presentation degree of the homology of a complex of FI-modules. This implies a linear stable range for the homology of congruence…
Let $p$ be a prime. In 1878 \'{E}. Lucas proved that the congruence $$ {p-1\choose k}\equiv (-1)^k\pmod{p}$$ holds for any nonnegative integer $k\in\{0,1,\ldots,p-1\}$. The converse statement was given in Problem 1494 of {\it Mathematics…
A logic has uniform interpolation if its formulas can be projected down to given subsignatures, preserving all logical consequences that do not mention the removed symbols; the weaker property of (Craig) interpolation allows the projected…
A classical result of A. Fleck states that if p is a prime, and n>0 and r are integers, then $$\sum_{k=r(mod p)}\binom {n}{k}(-1)^k=0 (mod p^{[(n-1)/(p-1)]}).$$ Recently R. M. Wilson used Fleck's congruence and Weisman's extension to…
In this paper, we see several basic properties of graded linear series. We firstly see that, if a graded linear series contains an ample series, then so are the pullbacks of the system under birational morphisms. Using this proposition, we…
We develop a topological approach to prove the generalized Lax conjecture using the fact that determinants of sufficiently big symmetric linear pencils are able to express the rigidly convex sets of RZ polynomials of any degree $d$.…
Generalized polynomials are mappings obtained from the conventional polynomials by the use of operations of addition, multiplication and taking the integer part. Extending the classical theorem of H. Weyl on equidistribution of polynomials,…
We introduce t-uniform simplicial complexes and we show that the lengths of spheres in such complexes are the terms of certain Lucas sequences. We find optimal constants for the linear isoperimetric inequality in the hyperbolic case.
We prove two congruences for the coefficients of power series expansions in t of modular forms where t is a modular function. As a result, we settle two recent conjectures of Chan, Cooper and Sica. Additionally, we provide a table of…
We generalize Luna's fundamental lemma to smooth morphisms between stacks with good moduli spaces. We also give a precise condition for when it holds for non-smooth morphisms and versions for coherent sheaves and complexes. This generalizes…
The Lucas sequences are integers defined by a homogeneous recurrence relation. They include the well-known Fibonacci numbers, which appear abundantly in nature. The complementary Lucas numbers, defined by the same recurrence relation, are…
A sharp estimation of the $L^p$-norms of some matrix coefficients of the square integrable representations is conjectured. The conjecture can be proved for integer values of $p$ using a result of J. Burbea.
Many generating series of combinatorially interesting numbers have the property that the sum of the terms of order $<p$ at some suitable point is congruent to a zero of a zeta-function modulo infinitely many primes $p$. Surprisingly, very…
We give a simple matrix-based proof of congruence equations modulo a prime $p$ involving sums of binomial coefficients appearing in Pascal's triangle. These equations can be used to construct some groups of exponent $p^n$. These groups, as…
Many integer sequences including the Catalan numbers, Motzkin numbers, and the Apr{\'e}y numbers can be expressed in the form ConstantTermOf$\left[P^nQ\right]$ for Laurent polynomials $P$ and $Q$. These are often called ``constant term…
In this article, we consider systems of linear congruences in several variables and obtain necessary and sufficient conditions as well as explicit expressions for the number of solutions subject to certain restriction conditions. These…
Motivated by the definition of the Gowers uniformity norms, we introduce and study a wide class of norms. Our aim is to establish them as a natural generalization of the $L_p$ norms. We shall prove that these normed spaces share many of the…
We consider the problem of existence of the diagonal representation for operators in the space of a family of generalized coherent states associated with an unitary irreducible representation of a (compact) Lie group. We show that necessary…
We define generalised equations of Z-Mahler type, based on the Zeckendorf numeration system. We show that if a sequence over a commutative ring is Z-regular, then it is the sequence of coefficients of a series which is a solution of a…