Related papers: Mod $p$ Buchstaber invariant

Let $p\equiv1\pmod 4$ be a prime. In this paper, with the help of Jacobsthal sums, we study some permutation problems involving biquadratic residues modulo $p$.

In this paper, we investigate the stabilizers of certain multisets $\mod p^k$ with respect to their natural multiplicative action, completely describing them for a certain family of polynomials whenever $p$ is an odd prime. This elucidates…

For prime $p\ge 7$, by using Baker's method we obtain two explicit bounds in terms of $p$ for the $j$-invariant of an integral point on $X_{\ns}^{+}(p)$ which is the modular curve of level $p$ corresponding to the normalizer of a non-split…

For modular indecomposable representations of a cyclic group $G$ of prime order $p$ we propose a list of polynomial invariants of degree $\leq 3$ that, together with a simple invariant of degree $p$, separate generic orbits and generate the…

We investigate the possibility to separate the bisimulation-invariant fragment of P from that of NP, resp. PSPACE. We build on Otto's Theorem stating that the bisimulation-invariant queries in P are exactly those that are definable in the…

We completely determine the mod $2$ Seiberg-Witten invariants for any spin structure on any closed, oriented, smooth $4$-manifold $X$. Our computation confirms the validity of the simple type conjecture mod $2$ for spin structures. Our…

New invariants for 2-dimensional cell complexes are defined, which can be interpreted as curvature bounds. These invariants are proved to be rational and computable in a companion article. This document is a survey that collects theorems…

Let p be an odd prime and let a be a positive integer. In this paper we investigate the sum $\sum_{k=0}^{p^a-1}\binom{hp^a-1}{k}\binom{2k}{k}/m^k$ mod p^2, where h,m are p-adic integers with m\not=0 (mod p). For example, we show that if…

Let $A$ be a finite or countable alphabet and let $\theta$ be literal (anti)morphism onto $A^*$ (by definition, such a correspondence is determinated by a permutation of the alphabet). This paper deals with sets which are invariant under…

Consider a complex one dimensional foliation on a complex surface near a singularity $p$. If $\mathcal{I}$ is a closed invariant set containing the singularity $p$, then $\mathcal{I}$ contains either a separatrix at $p$ or an invariant real…

Suppose that $f(x)=Ax^k$ mod $p$ is a permutation of the least residues mod $p$. With the exception of the maps $f(x)=Ax$ and $Ax^{(p+1)/2}$ mod $p$ we show that for fixed $n\geq 2$ the image of each residue class mod $n$ contains elements…

This paper is devoted to proving the differential invariance of the multiplicity of real and complex analytic sets. In particular, we prove the real version of Gau-Lipman's Theorem, i.e., it is proved that the multiplicity mod 2 of real…

Let p be a prime and let a be a positive integer. In this paper we determine $\sum_{k=0}^{p^a-1}\binom{2k}{k+d}/m^k$ and $\sum_{k=1}^{p-1}\binom{2k}{k+d}/(km^{k-1})$ modulo $p$ for all d=0,...,p^a, where m is any integer not divisible by p.…

In this paper, we show that for almost all primes p there is an integer solution x in [2,p-1] to the congruence x^x == x mod p. The solutions can be interpretated as fixed points of the map x -> x^x mod p, and we study numerically and…

Let $p>3$ be a prime number. We compute the rings of invariants of the elementary abelian $p$-group $(\mathbb Z/p\mathbb Z)^r$ for $3$-dimensional generic representations. Furthermore we show that these rings of invariants are complete…

We show that if p is an odd prime then $$\sum_{k=0}^{p-1}E_kE_{p-1-k}=1 (mod p)$$ and $$\sum_{k=0}^{p-3}E_kE_{p-3-k}=(-1)^{(p-1)/2}2E_{p-3} (mod p),$$ where E_0,E_1,E_2,... are Euler numbers. Moreover, we prove that for any positive integer…

Given a squarefree positive integer $d$, we want to find integers (or rational numbers with denominators not divisible by large primes) $a_0,a_1,a_2,\ldots$ such that for sufficiently large primes $p$ we have $\sum_{k=0}^{p-1}a_k\equiv…

Let $d(n)$ denote the number of divisors of $n$. In this paper, we study the average value of $d(a(p))$, where $p$ is a prime and $a(p)$ is the $p$-th Fourier coefficient of a normalized Hecke eigenform of weight $k \ge 2$ for $\Gamma_0(N)$…

We use modular invariant theory to establish a complete set of relations of the mod $p$ homology of $\{QS^k\}_{k\geq0}$, for $p$ odd, as a ring object in the category of coalgebras (also known as a coalgebraic ring or a Hopf ring). We also…

In the paper we treat Gale diagrams in a combinatorial way. The interpretation allows to describe simplicial complexes which are Alexander dual to boundaries of simplicial polytopes and, more generally, to nerve-complexes of general…