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We prove a version of a Nullstellensatz for partial exponential fields $(K,E)$, even though the ring of exponential polynomials $K[X_1,\ldots,X_n]^E$ is not a Hilbert ring. We show that under certain natural conditions one can embed an…

A systematic method is presented for the construction and classification of algebras of gauge transformations for arbitrary high rank tensor gauge fields. For every tensor gauge field of a given rank, the gauge transformation will be…

Steinberg's tensor product theorem shows that for semisimple algebraic groups the study of irreducible representations of higher Frobenius kernels reduces to the study of irreducible representations of the first Frobenius kernel. In the…

Let $H$ be a linear algebraic group over an algebraically closed field of characteristic $p>0$. We prove that any "exponential map" for $H$ induces a bijection between the variety of $r$-tuples of commuting $[p]$-nilpotent elements in…

In this paper we show that every finite-dimensional Zinbiel algebra over an arbitrary field is nilpotent, extending a previous result by other authors that they are solvable.

Let $\mathcal{K}:=(K;+,\cdot, D, 0, 1)$ be a differentially closed field of characteristic $0$ with field of constants $C$. In the first part of the paper we explore the connection between Ax-Schanuel type theorems (predimension…

Given a subfield $F$ of $\mathbb{C}$, we study the linear disjointess of the field $E$ generated by iterated exponentials of elements of $\overline{F}$, and the field $L$ generated by iterated logarithms, in the presence of Schanuel's…

Schinzel's Hypothesis H is a general conjecture in number theory on prime values of polynomials that generalizes, e.g., the twin prime conjecture and Dirichlet's theorem on primes in arithmetic progression. We prove an arithmetic analog of…

We develop geometry of algebraic subvarieties of $K^{n}$ over arbitrary Henselian valued fields $K$. This is a continuation of our previous article concerned with algebraic geometry over rank one valued fields. At the center of our approach…

We present an elementary method for proving enumeration formulas which are polynomials in certain parameters if others are fixed and factorize into distinct linear factors over Z. Roughly speaking the idea is to prove such formulas by…

We present constructive versions of Krull's dimension theory for commutative rings and distributive lattices. The foundations of these constructive versions are due to Joyal, Espa\~nol and the authors. We show that the notion of Krull…

Let X be an algebraic curve over Q and t a non-constant Q-rational function on X such that Q(t) is a proper subfield of Q(X). For every integer n pick a point P_n on X such that t(P_n)=n. We conjecture that, for large N, among the number…

In this paper we introduce the notion of existentially closed Leibniz algebras. Then we use HNN-extensions of Leibniz algebras in order to prove an embedding theorem.

We prove the Existential Closedness conjecture for the differential equation of the $j$-function and its derivatives. It states that in a differentially closed field certain equations involving the differential equation of the $j$-function…

Answering a long standing question, we give an example of a Hilbert module and a nonzero bounded right linear map having a kernel with trivial orthogonal complement. In particular, this kernel is different from its own double orthogonal…

Slopes of an adelic vector bundle exhibit a behaviour akin to successive minima. Comparisons between the two amount to a Siegel lemma. Here we use Zhang's version for absolute minima over the algebraic numbers. We prove a Minkowski-Hlawka…

Throughout the current paper, we extend the study of Zinbiel algebras to Zinbiel superalgebras. In particular, we show that all the Zinbiel superalgebras over an arbitrary field are nilpotent in the same way as occurs for Zinbiel algebras.…

In this paper, we consider real and complex algebras as well as algebras over general fields. In Section 2, we revisit and prove several results on (quadratic) algebras over general fields. As an example, we demonstrate that a quadratic…

A description of the Ziegler spectrum is given for trivial extensions of tubular algebras and related self-injective algebras of tubular type.

In this paper we define the algebraic sets and the ideal of points for bijective skew PBW extensions with coefficients in left Noetherian domains. Some properties of affine algebraic sets of commutative algebraic geometry will be extended,…