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Erhard Aichinger

Publications and source records attributed to Erhard Aichinger.

At least 19 recordsLinked to original sources
arXiv2026

Polynomial interpolation of partial functions in finite algebras with a Mal'cev term

We provide polynomial completeness results for finite algebras in congruence permutable varieties. In 2001, Idziak and S{\l}omczy{\'n}ska introduced the completeness concept of being \emph{polynomially rich}: a finite algebra is polynomially rich if every function preserving congruences and the Tame Congruence Theory labelling of prime quotients in the congruence lattice is a polynomial function of the algebra. We call a finite algebra \emph{strictly polynomially rich} if every partial congruence and type preserving function is a polynomial function, and we describe strictly polynomially rich algebras in congruence permutable varieties.

math.RA↗
arXiv2025

Structured and Punctured Nullstellens\"atze

A Nullstellensatz is a theorem providing information on polynomials that vanish on a certain set: David Hilbert's Nullstellensatz (1893) is a cornerstone of algebraic geometry, and Noga Alon's Combinatorial Nullstellensatz (1999) is a powerful tool in the "Polynomial Method", a technique used in combinatorics. Alon's Theorem excludes that a polynomial vanishing on a grid contains a monomial with certain properties. This theorem has been generalized in several directions, two of which we will consider in detail: Terence Tao and Van H. Vu (2006), Uwe Schauz (2008) and Micha\l{} Laso\'n (2010) exclude more monomials, and recently, Bogdan Nica (2023) improved the result for grids with additional symmetries in their side edges. Simeon Ball and Oriol Serra (2009) incorporated the multiplicity of zeros and gave Nullstellens\"atze for punctured grids, which are sets of the form $X \setminus Y$ with both $X,Y$ grids. We generalize some of these results; in particular, we provide a common generalization to the results of Schauz and Nica. To this end, we establish that during multivariate polynomial division, certain monomials are unaffected. This also allows us to generalize Pete L. Clark's proof of the nonzero counting theorem by Alon and F\"uredi to punctured grids.

math.CO↗
arXiv2025

On the complexity of solving equations over the symmetric group $S_4$

The complexity of solving equations over finite groups has been an active area of research over the last two decades, starting with Goldmann and Russell, \emph{The complexity of solving equations over finite groups} from 1999. One important case of a group with unknown complexity is the symmetric group $S_4.$ In 2023, Idziak, Kawa{\l}ek, and Krzaczkowski published $\exp(\Omega(\log^2 n))$ lower bounds for the satisfiability and equivalence problems over $S_4$ under the Exponential Time Hypothesis. In the present note, we prove that the satisfiability problem $\textsc{PolSat}(S_4)$ can be reduced to the equivalence problem $\textsc{PolEqv}(S_4)$ and thus, the two problems have the same complexity. We provide several equivalent formulations of the problem. In particular, we prove that $\textsc{PolEqv}(S_4)$ is equivalent to the circuit equivalence problem for $\operatorname{CC}[2,3,2]$-circuits, which were introduced by Idziak, Kawe{\l}ek and Krzaczkowski. Under their strong exponential size hypothesis, such circuits cannot compute $\operatorname{AND}_n$ in size $\exp(o(\sqrt{n})).$ Our results provide an upper bound on the complexity of $\textsc{PolEqv}(S_4)$ that is based on the minimal size of $\operatorname{AND}_n$ over $\operatorname{CC}[2,3,2]$-circuits.

cs.CC↗
arXiv2023

On when the union of two algebraic sets is algebraic

In universal algebraic geometry, an algebra is called an equational domain if the union of two algebraic sets is algebraic. We characterize equational domains, with respect to polynomial equations, inside congruence permutable varieties, and with respect to term equations, among all algebras of size two and all algebras of size three with a cyclic automorphism. Furthermore, for each size at least three, we prove that, modulo term equivalence, there is a continuum of equational domains of that size.

math.RA↗
arXiv2023

Zero testing and equation solving for sparse polynomials on rectangular domains

We consider sparse polynomials in $N$ variables over a finite field, and ask whether they vanish on a set $S^N$, where $S$ is a set of nonzero elements of the field. We see that if for a polynomial $f$, there is $\mathbf{c}\in S^N$ with $f (\mathbf{c}) \neq 0$, then there is such a $\mathbf{c}$ in every sphere inside $S^N$, where the radius of the sphere is bounded by a multiple of the logarithm of the number of monomials that appear in $f$. A similar result holds for the solutions of the equations $f_1 = \cdots = f_r = 0$ inside $S^N$.

math.RA↗
arXiv2022

Finite representation of commutator sequences

Several structural properties of a universal algebra can be seen from the higher commutators of its congruences. Even on a finite algebra, the sequence of higher commutator operations is an infinite object. In the present paper, we exhibit finite representations of this sequence.

math.RA↗
arXiv2022

On the number of universal algebraic geometries

The algebraic geometry of a universal algebra $\mathbf{A}$ is defined as the collection of solution sets of term equations. Two algebras $\mathbf{A}_1$ and $\mathbf{A}_2$ are called algebraically equivalent if they have the same algebraic geometry. We prove that on a finite set $A$ with $\lvert A \rvert >3$ there are countably many algebraically inequivalent Mal'cev algebras and that on a finite set $A$ with $\lvert A \rvert >2$ there are continuously many algebraically inequivalent algebras.

math.RA↗
arXiv2019

Chevalley Warning type results on abelian groups

We develop a notion of degree for functions between two abelian groups that allows us to generalize the Chevalley Warning Theorems from fields to noncommutative rings or abelian groups of prime power order.

math.GR↗
arXiv2019

A clonoid based approach to some finiteness results in universal algebraic geometry

We prove that for a finite first order structure $\mathbf{A}$ and a set of first order formulas $\Phi$ in its language with certain closure properties, the finitary relations on $A$ that are definable via formulas in $\Phi$ are uniquely determined by those of arity $|A|^{2}$. This yields new proofs for some finiteness results from universal algebraic geometry.

math.LO↗
arXiv2019

Bounding the free spectrum of nilpotent algebras of prime power order

Let $\mathbf{A}$ be a finite nilpotent algebra in a congruence modular variety with finitely many fundamental operations. If $\mathbf{A}$ is of prime power order, then it is known that there is a polynomial $p$ such that for every $n \in \mathbb{N}$, every $n$-generated algebra in the variety generated by $\mathbf{A}$ has at most $2^{p(n)}$ elements. We present a bound on the degree of this polynomial.

math.RA↗
arXiv2019

Solving systems of equations in supernilpotent algebras

Recently, M. Kompatscher proved that for each finite supernilpotent algebra $\mathbf{A}$ in a congruence modular variety, there is a polynomial time algorithm to solve polynomial equations over this algebra. Let $μ$ be the maximal arity of the fundamental operations of $\mathbf{A}$, and let \[ d := |A|^{\log_2 (μ) + \log_2 (|A|) + 1}.\] Applying a method that G. Károlyi and C. Szabó had used to solve equations over finite nilpotent rings, we show that for $\mathbf{A}$, there is $c \in \mathbb{N}$ such that a solution of every system of $s$ equations in $n$ variables can be found by testing at most $c n^{sd}$ (instead of all $|A|^n$ possible) assignments to the variables. This also yields new information on some circuit satisfiability problems.

math.LO↗
arXiv2017

Complexity of term representations of finitary functions

The clone of term operations of an algebraic structure consists of all operations that can be expressed by a term in the language of the structure. We consider bounds for the length and the height of the terms expressing these functions, and we show that these bounds are often robust against the change of the basic operations of the structure.

math.RA↗
arXiv2017

Subnearrings of $(\mathbb{Z}[x],+,\circ)$

We show that the nearring $(\mathbb{Z}[x],+,\circ)$ of integer polynomials, where the nearring multiplication is the composition of polynomials, has uncountably many subnearrings, and we give an explicit description of those nearrings that are generated by subsets of $\{1,x,x^2,x^3\}$.

math.RA↗
arXiv2016

Congruence lattices forcing nilpotency

Given a lattice $\mathbb{L}$ and a class $K$ of algebraic structures, we say that $\mathbb{L}$ \emph{forces nilpotency} in $K$ if every algebra $\mathbf{A} \in K$ whose congruence lattice $\mathrm{Con} (\mathbf{A})$ is isomorphic to $\mathbb{L}$ is nilpotent. We describe congruence lattices that force nilpotency, supernilpotency or solvability for some classes of algebras. For this purpose, we investigate which commutator operations can exist on a given congruence lattice.

math.RA↗
arXiv2016

On the local closure of clones on countable sets

We consider clones on countable sets. If such a clone has quasigroup operations, is locally closed and countable, then there is a function $f : \mathbb{N} \to \mathbb{N}$ such that the $n$-ary part of $C$ is equal to the $n$-ary part of $\mathrm{Pol}\,\mathrm{Inv}^{[f(n)]} C$, where $\mathrm{Inv}^{[f(n)]} C$ denotes the set of $f(n)$-ary invariant relations of $C$.

math.LO↗