Related papers: Schinzel's Problem: Imprimitive covers and the mon…

A solution $(x,y,z) \in \mathbb{Z}^3-\{(0,0,0)\}$ to a generalized Fermat equation \[ Ax^a + By^b + Cz^c = 0, \] is called \emph{primitive} if $\gcd(x,y,z) = 1$. By work of Beukers, we know that in the \emph{spherical} regime (that is, when…

This diploma thesis is concerned with functional decomposition $f = g \circ h$ of polynomials. First an algorithm is described which computes decompositions in polynomial time. This algorithm was originally proposed by Zippel (1991). A…

A set of integers is primitive if no number in the set divides another. We introduce a new method for bounding Erd\H{o}s sums of primitive sets, suggested from output of GPT-5.4 Pro, based on Markov chains with von Mangoldt weights. The…

We show that the Schur polynomials in all primitive $n$th roots of unity are $1$, $0$, or $-1$, if $n$ has at most two distinct odd prime factors. This result can be regarded as a generalization of properties of the coefficients of the…

Let $F=\mathbb{F}_{q^m}$, $m>6$, $n$ a positive integer, and $f=p/q$ with $p$, $q$ co-prime irreducible polynomials in $F[x]$ and deg$(p)$ $+$ deg$(q)= n$. A sufficient condition has been obtained for the existence of primitive pairs…

The Gan-Gross-Prasad problem is to describe the restriction of representations of a classical group $G$ to smaller groups $H$ of the same kind. In this paper, we solved the Gan-Gross-Prasad problem over finite fields completely. In previous…

The monodromy conjecture is a mysterious open problem in singularity theory. Its original version relates arithmetic and topological/geometric properties of a multivariate polynomial $f$ over the integers, more precisely, poles of the…

This paper deals with differential pencils possessing a term depending on the unknown function with a fixed argument. We deduce the so called main equation together with its fine structure for the spectral problem. Then, according to the…

Matrix pencils, or pairs of matrices, may be used in a variety of applications. In particular, a pair of matrices (E,A) may be interpreted as the differential equation E x' + A x = 0. Such an equation is invariant by changes of variables,…

Given two polynomials $P(\underline x)$, $Q(\underline x)$ in one or more variables and with integer coefficients, how does the property that they are coprime relate to their values $P(\underline n), Q(\underline n)$ at integer points…

We study conditions on a $C^*$-dynamical system $(A,G,\alpha)$ under which induction of primitive ideals (resp. irreducible representations) from stabilizers for the action of $G$ on the primitive ideal space $\Prim(A)$ give primitive…

The original Riemann-Hilbert problem asks to find a Fuchsian ordinary differential equation with prescribed singularities and monodromy in the complex line. In the early 1980's Kashiwara solved a generalized version of the problem, valid on…

The splice quotients are an interesting class of normal surface singularities with rational homology sphere links, defined by W. Neumann and J. Wahl. If Gamma is a tree of rational curves that satisfies certain combinatorial conditions,…

In this paper we study complex representations of the factorpower $\fp(G,M)$ of a finite group $G$ acting on a finite set $M$. This includes the finite monoid $\FP$, which can be seen as a kind of a ``balanced'' generalization of the…

Properly specializing the parameters in ``Schnizer modules'', for type A,B,C and D, we get its unique primitive vector. Then we show that the module generated by the primitive vector is an irreducible highest weight module of finite…

Using the properties of the ideal of the coordinate Hermite interpolation on n-dimensional grid [4], we prove that the extension k in k[x1, x2, ..., xn] / (f1(x1), ..., fn(xn)) has a primitive element if and only if at most one of the…

Our aim is to find a general approach to the theory of classical solutions of the Garnier system in $n$-variables, ${\cal G}_n$, based on the Riemann-Hilbert problem and on the geometry of the space of isomonodromy deformations. Our…

The Schubert varieties on a flag manifold G/P give rise to a cell decomposition on G/P whose Kronecker duals, known as the Schubert classes on G/P, form an additive base of the integral cohomology of G/P. The Schubert's problem of…

The Schinzel hypothesis is a famous conjectural statement about primes in value sets of polynomials, which generalizes the Dirichlet theorem about primes in an arithmetic progression. We consider the situation that the ring of integers is…

Let $G$ be a finite group, and $F$ a field of characteristic $0$ or prime to the order of $G$. In $1952$, Witt and in $1956$, Berman independently proved that the number of inequivalent irreducible $F$-representations of $G$ is equal to the…