Related papers: Equations in the Hadamard ring of rational functio…

Let $I$ be a homogeneous ideal in the polynomial ring $R = k[z_1, \cdots, z_n]$ , where $k$ is an algebraically closed field of characteristic zero. Macaulay's Theorem provides constraints on the Hilbert function of $I$ or $R/I$ from one…

Take a multiplicative monoid of sequences in which the multiplication is given by Hadamard product. The set of linear combinations of interleaving monoid elements then yields a ring. For hypergeometric sequences, the resulting ring is a…

We discuss two conjectures. (I) For each x_1,...,x_n \in R (C) there exist y_1,...,y_n \in R (C) such that \forall i \in {1,...,n} |y_i| \leq 2^{2^{n-2}} \forall i \in {1,...,n} (x_i=1 \Rightarrow y_i=1) \forall i,j,k \in {1,...,n}…

Suppose that 2d-2 tangent lines to the rational normal curve z\mapsto (1 : z : ... : z^d) in d-dimensional complex projective space are given. It was known that the number of codimension 2 subspaces intersecting all these lines is always…

Let $G$ be a finite group and $k$ be a field. Let $G$ act on the rational function field $k(x_g:g\in G)$ by $k$-automorphisms defined by $g\cdot x_h=x_{gh}$ for any $g,h\in G$. Noether's problem asks whether the fixed field $k(G)=k(x_g:g\in…

Answering a question of J.~Kovacic, we show that, for any Keigher ring, its differential spectrum coincides with the differential spectrum of the ring of global sections of the structure sheaf. In particular, we obtain the answer for Ritt…

We classify all rational maps $H \in K(x)^n$ for which ${\rm trdeg}_K K(tH_1,tH_2,\ldots,tH_n) \le 2$, where $K$ is any field and $t$ is another indeterminate. Furthermore, we classify all such maps for which additionally $JH \cdot H = {\rm…

The ring operations and the metric on $C(X)$ are extended to the set $\mathbb{H}_{nf}(X)$ of all nearly finite Hausdorff continuous interval valued functions and it is shown that $\mathbb{H}_{nf}(X)$ is both rationally and topologically…

The Euclidean algorithm makes possible a simple but powerful generalization of Taylor's theorem. Instead of expanding a function in a series around a single point, one spreads out the spectrum to include any number of points with given…

In a previous paper by the author a universal ring of invariants for algebraic structures of a given type was constructed. This ring is a polynomial algebra that is generated by certain trace diagrams. It was shown that this ring admits the…

We link the QUMOND theory with the Helmholtz-Weyl decomposition and introduce a new formula for the gradient of the Mondian potential using singular integral operators. This approach allows us to demonstrate that, under very general…

Let $R\subset F$ be an extension of real closed fields and ${\mathcal S}(M,R)$ the ring of (continuous) semialgebraic functions on a semialgebraic set $M\subset R^n$. We prove that every $R$-homomorphism $\varphi:{\mathcal S}(M,R)\to F$ is…

In this paper we present algorithmic considerations and theoretical results about the relation between the orders of certain groups associated to the components of a polynomial and the order of the group that corresponds to the polynomial,…

We study rational homology groups of one-point compactifications of spaces of complex monic polynomials with multiple roots. These spaces are indexed by number partitions. A standard reformulation in terms of quotients of orbit arrangements…

In this paper we give a factorization theorem for the ring of exponential polynomials in many variables over an algebraically closed field of characteristic 0 with an exponentiation. This is a generalization of the factorization theorem due…

Let $k$ be a number field with cyclotomic closure $k^{\mathrm{cyc}}$, and let $h \in k^{\mathrm{cyc}}(x)$. For $A \ge 1$ a real number, we show that \[ \{ \alpha \in k^{\mathrm{cyc}} : h(\alpha) \in \overline{\mathbb Z} \text{ has house at…

Let f(t,X) be an irreducible polynomial over the field of rational functions k(t), where k is a number field. Let O be the ring of integers of k. Hilbert's irreducibility theorem gives infinitely many integral specializations of t to values…

We show that Mazur's conjecture on the real topology of rational points on varieties implies that there is no diophantine model of the rational integers in the rational numbers. We also prove that there is a diophantine model of the…

For a convex polytope P with rational vertices, we count the number of integer points in integral dilates of P and its interior. The Ehrhart-Macdonald reciprocity law gives an intimate relation between these two counting functions. A…

A compact subset $K$ of the complex plane $\C$ is a set of polynomial (respectively rational) approximation if $P(K)=A(K)$ (respectively $R(K)=A(K)$), where $P(K)$ (respectively $R(K)$) is the family of functions on $K$ which are uniform…