Related papers: Construction of ray class fields by smaller genera…

We introduce a new type of reduction of inversive difference polynomials that is associated with a partition of the basic set of automorphisms $\sigma$ and uses a generalization of the concept of effective order of a difference polynomial.…

We determine the most general group of equivalence transformations for a family of differential equations defined by an arbitrary vector field on a manifold. We also find all invariants and differential invariants for this group up to the…

Using the Riemann Hypothesis over finite fields and bounds for the size of spherical codes, we give explicit upper bounds, of polynomial size with respect to the size of the field, for the number of geometric isomorphism classes of…

A minimal (by inclusion) generating set for the algebra of semi-invariants of a quiver of dimension (2,...,2) is established over an infinite field of arbitrary characteristic. The mentioned generating set consists of the determinants of…

One of basic difficulties of machine learning is handling unknown rotations of objects, for example in image recognition. A related problem is evaluation of similarity of shapes, for example of two chemical molecules, for which direct…

Constructing $r$-th nonresidue over a finite field is a fundamental computational problem. A related problem is to construct an irreducible polynomial of degree $r^e$ (where $r$ is a prime) over a given finite field $\mathbb{F}_q$ of…

In this note is we exhibit an elementary method to construct explicitly curves over finite fields with many points. Despite its elementary character the method is very efficient and can be regarded as a partial substitute for the use of…

Let $\ell$ and $p$ be odd primes. For a positive integer $\mu$ let $k_\mu$ be the ray class field of $k=\mathbb{Q}(e^{2\pi i/\ell})$ modulo $2p^\mu$. We present certain class fields $K_\mu$ of $k$ such that $k_\mu\leq K_\mu\leq k_{\mu+1}$,…

We construct two-dimensional non-commutative topological quantum field theories (TQFTs), one for each Hecke algebra corresponding to a finite Coxeter system. These TQFTs associate an invariant to each ciliated surface, which is a Laurent…

We discuss two simple but useful observations that allow the construction of modular forms from given ones using invariant theory. The first one deals with elliptic modular forms and their derivatives, and generalizes the Rankin-Cohen…

We determine all isomorphism classes of hyperfields of a given finite order which can be obtained as quotients of finite fields of sufficiently large order. Using this result, we determine which hyperfields of order at most 4 are quotients…

We propose the method for obtaining invariants of arbitrary representations of Lie groups that reduces this problem to known problems of linear algebra. The basis of this method is the idea of a special extension of the representation…

Higher genus partition functions of two-dimensional conformal field theories have to be invariants under linear actions of mapping class groups. We illustrate recent results [4,6] on the construction of such invariants by concrete…

We give a brief re-exposition of the theory due to Pauli and Sinclair of ramification polygons of Eisenstein polynomials over p-adic fields, their associated residual polynomials and an algorithm to produce all extensions for a given…

A new class of 3-manifold invariants is constructed from representations of the category of framed tangles.

In the making of origami, one starts with a piece of paper, and through a series of folds along seed points one constructs complicated three-dimensional shapes. Mathematically, one can think of the complex numbers as representing the piece…

We introduce a zeta function counting imaginary quadratic number fields by their class numbers. It is proved that such a function is rational depending only on the eight roots of unity of degrees $1$ and $2$. As a corollary, one gets a…

In this note, we consider applications of Ratner's theorem to constructions of families of polynomials with dense values on the set of primitive integer points from the viewpoint of invariant theory.

We use quilted Floer theory to construct functor-valued invariants of tangles arising from moduli spaces of flat bundles on punctured surfaces. As an application, we show the non-triviality of certain elements in the symplectic mapping…

We consider the classical problem of invariant generation for programs with polynomial assignments and focus on synthesizing invariants that are a conjunction of strict polynomial inequalities. We present a sound and semi-complete method…