arXiv subjects

Martin Kreuzer

Publications and source records attributed to Martin Kreuzer.

At least 19 recordsLinked to original sources

arXiv2026

Graded Algebras over Polynomial Rings

Given a trivially graded polynomial ring $A=K[a_1,\dots,a_m]$ over a field $K$ and a positively graded polynomial ring $P=A[x_1,\dots,x_k]$, we study graded rings $R=P/I$, where $I$ is a homogeneous ideal in $P$ such that $I\cap A = \{0\}$. The corresponding morphism $\Theta: {\rm Spec}(R) \rightarrow {\rm Spec}(A) = \mathbb{A}^m_K$ is used to prove that ${\rm Spec}(R)$ is connected. Then we characterize and compute the following loci in $\mathbb{A}^m_K$: the set ${\rm Sing}_0(\Theta)$ of all points such that the corresponding point in the zero section of $\Theta$ is singular in ${\rm Spec}(R)$, the set ${\rm Sing}_v(\Theta)$ of all points $\Gamma$ such that the origin of the fiber $F_\Gamma$ of $\Theta$ is singular, and the set ${\rm Sing}_s(\Theta)$ of all points $\Gamma$ such that $\dim({\rm Sing}(F_\Gamma)) \ge 1$. These results are then used to study MaxDeg border basis schemes, as their coordinate rings are non-negatively graded by the total arrow degree and they have the required structure. In particular, we explicitly determine the singular loci for the $\mathcal{O}$-border basis schemes with $\mathcal{O}=\{1,x,y,z,z^2\}$ and $\mathcal{O} = \{1,x,y,z,yz\}$.

math.AC↗

arXiv2025

Subgroups of Cyclically Amalgamated Free Products

Given a group $G = H_1 \ast_A H_2$ which is the free product of two finitely generated groups $H_1$ and $H_2$ with amalgamation over a cyclic subgroup $A$ which is malnormal in $G$, we study relations between the structure of its subgroups and the structure of the group $G$ itself. Firstly, we show that if $H_1$ and $H_2$ are 3-free products of cyclics of rank $\ge 3$ then $G$ is also a 3-free product of cyclics. Secondly, we prove that if $H_1$ and $H_2$ are 4-free products of cyclics of rank $\ge 4$ then every 4-generated subgroup of $G$ is a free product of $\le 4$ cyclics or a 1-relator quotient of a free product of four cyclic groups. Here a group is called an $n$-free product of cyclics if every $n$-generated subgroup is a free product of $\le n$ cyclic groups. These results are based on ubiquitous applications of the Nielsen method for amalgamated free products which we recall carefully. Lastly, given an infinite, finitely presented group which is not free, but all of its infinite index subgroups are free, a well-known conjecture says that it is isomorphic to a surface group. We revisit and elaborate on predominantly group theoretic proofs of this conjecture for cyclically amalgamated products as above, as well as for certain HNN extensions.

math.GR↗

arXiv2025

Code Equivalence, Point Set Equivalence, and Polynomial Isomorphism

The linear code equivalence (LCE) problem is shown to be equivalent to the point set equivalence (PSE) problem, i.e., the problem to check whether two sets of points in a projective space over a finite field differ by a linear change of coordinates. For such a point set $\mathbb{X}$, let $R$ be its homogeneous coordinate ring and $\mathfrak{J}_{\mathbb{X}}$ its canonical ideal. Then the LCE problem is shown to be equivalent to an algebra isomorphism problem for the doubling $R/\mathfrak{J}_{\mathbb{X}}$. As this doubling is an Artinian Gorenstein algebra, we can use its Macaulay inverse system to reduce the LCE problem to a Polynomial Isomorphism (PI) problem for homogeneous polynomials. The last step is polynomial time under some mild assumptions about the codes. Moreover, for indecomposable iso-dual codes we can reduce the LCE search problem to the PI search problem of degree 3 by noting that the corresponding point sets are self-associated and arithmetically Gorenstein, so that we can use the isomorphism problem for the Artinian reductions of the coordinate rings and form their Macaulay inverse systems.

cs.IT↗

arXiv2025

Trace Minimization and Roots in ${\rm PSL}(2,\mathbb{R})$

Suppose that $A,B \in {\rm PSL}(2,\mathbb{R})$ generate a non-elementary Fuchsian group. Let $m,n\in\mathbb{N}_+$, and let $R,S\in {\rm PSL}(2,\mathbb{R})$ such that $R^m=A$ and $S^n=B$. We present explicit algorithms to check whether $\langle R,S\rangle$ is a Fuchsian group. These algorithms rely only on the knowledge of the traces ${\rm tr}(A)$, ${\rm tr}(B)$, and ${\rm tr}(AB)$, which we assume to be given as algebraic numbers. The main tools are the classic Trace Minimization Algorithm, as introduced in 1972 by the third author, a new Extended Trace Minimization Algorithm, and a Rational Angle Recovery Algorithm which checks whether a given number $x$ is if the form $x = 2 \cos(p \pi/q)$. The question when roots of the generators of a free Fuchsian group of rank 2 generate again a free Fuchsian group of rank 2, and an extension to positive rational exponents $m,n$ are treated, as well.

math.GR↗

arXiv2025

Re-Embeddings of Special Border Basis Schemes

Border basis schemes are open subschemes of the Hilbert scheme of $\mu$ points in an affine space $\mathbb{A}^n$. They have easily describable systems of generators of their vanishing ideals for a natural embedding into a large affine space $\mathbb{A}^{\mu\nu}$. Here we bring together several techniques for re-embedding affine schemes into lower dimensional spaces which we developed in the last years. We study their efficacy for some special types of border basis schemes such as MaxDeg border basis schemes, L-shape and simplicial border basis schemes, as well as planar border basis schemes. A particular care is taken to make these re-embeddings efficiently computable and to check when we actually get an isomorphism with $\mathbb{A}^{n\mu}$, i.e., when the border basis scheme is an affine cell.

math.AG↗

arXiv2024

Efficiently Checking Separating Indeterminates

In this paper we continue the development of a new technique for computing elimination ideals by substitution which has been called $Z$-separating re-embeddings. Given an ideal $I$ in the polynomial ring $K[x_1,\dots,x_n]$ over a field $K$, this method searches for tuples $Z=(z_1,\dots,z_s)$ of indeterminates with the property that $I$ contains polynomials of the form $f_i = z_i - h_i$ for $i=1,\dots,s$ such that no term in $h_i$ is divisible by an indeterminate in $Z$. As there are frequently many candidate tuples $Z$, the task addressed by this paper is to efficiently check whether a given tuple $Z$ has this property. We construct fast algorithms which check whether the vector space spanned by the generators of $I$ or a somewhat enlarged vector space contain the desired polynomials $f_i$. We also extend these algorithms to Boolean polynomials and apply them to cryptoanalyse round reduced versions of the AES cryptosystem faster.

math.AC↗

arXiv2024

Computing the unit group of a commutative finite $\mathbb{Z}$-algebra

For a commutative finite $\mathbb{Z}$-algebra, i.e., for a commutative ring $R$ whose additive group is finitely generated, it is known that the group of units of $R$ is finitely generated, as well. Our main results are algorithms to compute generators and the structure of this group. This is achieved by reducing the task first to the case of reduced rings, then to torsion-free reduced rings, and finally to an order in a reduced ring. The simplified cases are treated via a calculation of exponent lattices and various algorithms to compute the minimal primes, primitive idempotents, and other basic objects. All algorithms have been implemented and are available as a SageMath package. Whenever possible, the time complexity of the described methods is tracked carefully.

math.AC↗

arXiv2024

Efficient Algorithms for Finite $\mathbb{Z}$-Algebras

For a finite $\mathbb{Z}$-algebra $R$, i.e., for a $\mathbb{Z}$-algebra which is a finitely generated $\mathbb{Z}$-module, we assume that $R$ is explicitly given by a system of $\mathbb{Z}$-module generators $G$, its relation module ${\rm Syz}(G)$, and the structure constants of the multiplication in $R$. In this setting we develop and analyze efficient algorithms for computing essential information about $R$. First we provide polynomial time algorithms for solving linear systems of equations over $R$ and for basic ideal-theoretic operations in $R$. Then we develop ZPP (zero-error probabilitic polynomial time) algorithms to compute the nilradical and the maximal ideals of 0-dimensional affine algebras $K[x_1,\dots,x_n]/I$ with $K=\mathbb{Q}$ or $K=\mathbb{F}_p$. The task of finding the associated primes of a finite $\mathbb{Z}$-algebra $R$ is reduced to these cases and solved in ZPPIF (ZPP plus one integer factorization). With the same complexity, we calculate the connected components of the set of minimal associated primes ${\rm minPrimes}(R)$ and then the primitive idempotents of $R$. Finally, we prove that knowing an explicit representation of $R$ is polynomial time equivalent to knowing a strong Gröbner basis of an ideal $I$ such that $R = \mathbb{Z}[x_1,\dots,x_n]/I$.

math.AC↗

arXiv2024

Elimination by Substitution

Let $K$ be a field and $P=K[x_1,\dots,x_n]$. The technique of elimination by substitution is based on discovering a coherently $Z=(z_1,\dots,z_s)$-separating tuple of polynomials $(f_1,\dots,f_s)$ in an ideal $I$, i.e., on finding polynomials such that $f_i = z_i - h_i$ with $h_i \in K[X \setminus Z]$. Here we elaborate on this technique in the case when $P$ is non-negatively graded. The existence of a coherently $Z$-separating tuple is reduced to solving several $P_0$-module membership problems. Best separable re-embeddings, i.e., isomorphisms $P/I \longrightarrow K[X \setminus Z] / (I \cap K[X \setminus Z])$ with maximal $\#Z$, are found degree-by-degree. They turn out to yield optimal re-embeddings in the positively graded case. Viewing $P_0 \longrightarrow P/I$ as a fibration over an affine space, we show that its fibers allow optimal $Z$-separating re-embeddings, and we provide a criterion for a fiber to be isomorphic to an affine space. In the last section we introduce a new technique based on the solution of a unimodular matrix problem which enables us to construct automorphisms of $P$ such that additional $Z$-separating re-embeddings are possible. One of the main outcomes is an algorithm which allows us to explicitly compute a homogeneous isomorphism between $P/I$ and a non-negatively graded polynomial ring if $P/I$ is regular.

math.AC↗

arXiv2024

Re-embeddings of Affine Algebras Via Gröbner Fans of Linear Ideals

Given an affine algebra $R=K[x_1,\dots,x_n]/I$ over a field $K$, where $I$ is an ideal in the polynomial ring $P=K[x_1,\dots,x_n]$, we examine the task of effectively calculating re-embeddings of $I$, i.e., of presentations $R=P'/I'$ such that $P'=K[y_1,\dots,y_m]$ has fewer indeterminates. For cases when the number of indeterminates $n$ is large and Gröbner basis computations are infeasible, we have previously introduced the method of $Z$-separating re-embeddings. This method tries to detect polynomials of a special shape in $I$ which allow us to eliminate the indeterminates in the tuple $Z$ by a simple substitution process. Here we improve this approach by showing that suitable candidate tuples $Z$ can be found using the Gröbner fan of the linear part of $I$. Then we describe a method to compute the Gröbner fan of a linear ideal, and we improve this computation in the case of binomial linear ideals using a cotangent equivalence relation. Finally, we apply the improved technique in the case of the defining ideals of border basis schemes.

math.AC↗

arXiv2023

Optimal Re-Embeddings of Border Basis Schemes

Border basis schemes are open subschemes of Hilbert schemes parametrizing 0-dimensional subschemes of $\mathbb{P}^n$ of given length. They yield open coverings and are easy to describe and to compute with. Our topic is to find re-embeddings of border basis schemes into affine spaces of minimal dimension. Given $P = K[X] = K[x_1,\dots,x_n]$, an ideal $I\subseteq \langle X \rangle$, and a tuple $Z$ of indeterminates, in previous papers the authors developed techniques for computing $Z$-separating re-embeddings of $I$, i.e., of isomorphisms $Φ: P/I \rightarrow K[X\setminus Z] / (I\cap K[X\setminus Z])$. Here these general techniques are developed further and improved by constructing a new algorithm for checking candidate tuples $Z$ and by using the Gröbner fan of the linear part of $I$ advantageously. Then we apply this to the ideals defining border basis schemes $\mathbb{B}_{\mathcal{O}}$, where $\mathcal{O}$ is an order ideal of terms, and to their natural generating polynomials. The fact that these ideals are homogeneous w.r.t. the arrow grading allows us to look for suitable tuples $Z$ more systematically. Using the equivalence of indeterminates modulo the square of the maximal ideal, we compute the Gröbner fan of the linear part of the ideal quickly and determine which indeterminates should be in $Z$ when we are looking for optimal re-embeddings. Specific applications include re-embeddings of border basis schemes where $\mathcal{O}\subseteq K[x,y]$ and where $\mathcal{O}$ consists of all terms up to some degree.

math.AG↗

arXiv2023

SAT Solving Using XOR-OR-AND Normal Forms

This paper introduces the XOR-OR-AND normal form (XNF) for logical formulas. It is a generalization of the well-known Conjunctive Normal Form (CNF) where literals are replaced by XORs of literals. As a first theoretic result, we show that every CNF formula is equisatisfiable to a formula in 2-XNF, i.e., a formula in XNF where each clause involves at most two XORs of literals. Subsequently, we present an algorithm which converts Boolean polynomials efficiently from their Algebraic Normal Form (ANF) to formulas in 2-XNF. Experiments with the cipher ASCON-128 show that cryptographic problems, which by design are based strongly on XOR-operations, can be represented using far fewer variables and clauses in 2-XNF than in CNF. In order to take advantage of this compact representation, new SAT solvers based on input formulas in 2-XNF need to be designed. By taking inspiration from graph-based 2-CNF SAT solving, we devise a new DPLL-based SAT solver for formulas in 2-XNF. Among others, we present advanced pre- and in-processing techniques. Finally, we give timings for random 2-XNF instances and instances related to key recovery attacks on round reduced ASCON-128, where our solver outperforms state-of-the-art alternative solving approaches.

cs.LO↗

arXiv2023

Decomposing Finite $\mathbb{Z}$-Algebras

For a finite $\mathbb{Z}$-algebra $R$, i.e., for a ring which is not necessarily associative or unitary, but whose additive group is finitely generated, we construct a decomposition of $R/{\rm Ann}(R)$ into directly indecomposable factors under weak hypotheses. The method is based on constructing and decomposing a ring of scalars $S$, and then lifting the decomposition of $S$ to the bilinear map given by the multiplication of $R$, and finally to $R/{\rm Ann}(R)$. All steps of the construction are given as explicit algorithms and it is shown that the entire procedure has a probabilistic polynomial time complexity in the bit size of the input, except for the possible need to calculate the prime factorization of one integer. In particular, in the case when ${\rm Ann}(R) = 0$, these algorithms compute a direct decomposition of $R$ into directly indecomposable factors.

math.RA↗

arXiv2023

Computing the Binomial Part of a Polynomial Ideal

Given an ideal $I$ in a polynomial ring $K[x_1,\dots,x_n]$ over a field $K$, we present a complete algorithm to compute the binomial part of $I$, i.e., the subideal ${\rm Bin}(I)$ of $I$ generated by all monomials and binomials in $I$. This is achieved step-by-step. First we collect and extend several algorithms for computing exponent lattices in different kinds of fields. Then we generalize them to compute exponent lattices of units in 0-dimensional $K$-algebras, where we have to generalize the computation of the separable part of an algebra to non-perfect fields in characteristic $p$. Next we examine the computation of unit lattices in affine $K$-algebras, as well as their associated characters and lattice ideals. This allows us to calculate ${\rm Bin}(I)$ when $I$ is saturated with respect to the indeterminates by reducing the task to the 0-dimensional case. Finally, we treat the computation of ${\rm Bin}(I)$ for general ideals by computing their cellular decomposition and dealing with finitely many special ideals called $(s,t)$-binomial parts. All algorithms have been implemented in SageMath.

math.AC↗

arXiv2023

Differential theory of zero-dimensional schemes

For a 0-dimensional scheme $\mathbb{X}$ in $\mathbb{P}^n$ over a perfect field $K$, we first embed the homogeneous coordinate ring $R$ into its truncated integral closure $\widetilde{R}$. Then we use the corresponding map from the module of Kähler differentials $Ω^1_{R/K}$ to $Ω^1_{\widetilde{R}/K}$ to find a formula for the Hilbert polynomial ${\rm HP}(Ω^1_{R/K})$ and a sharp bound for the regularity index ${\rm ri}(Ω^1_{R/K})$. Additionally, we extend this to formulas for the Hilbert polynomials ${\rm HP}(Ω^m_{R/K})$ and bounds for the regularity indices of the higher modules of Kähler differentials. Next we derive a new characterization of a weakly curvilinear scheme $\mathbb{X}$ which can be checked without computing a primary decomposition of its homogeneous vanishing ideal. Moreover, we prove precise formulas for the Hilbert polynomial of $Ω^m_{R/K}$ of a fat point scheme $\mathbb{X}$, extending and settling previous partial results and conjectures. Finally, we characterize uniformity conditions on $\mathbb{X}$ using the Hilbert functions of the Kähler differential modules of $\mathbb{X}$ and its subschemes.

math.AC↗

arXiv2022

Restricted Gröbner fans and re-embeddings of affine algebras

In this paper we continue the study of good re-embeddings of affine K-algebras started in [KLR]. The idea is to use special linear projections to find isomorphisms between a given affine K-algebra K[X]/I, where X=(x_1,...,x_n), and K-algebras having fewer generators. These projections are induced by particular tuples of indeterminates Z and by term orderings $σ$ which realize Z as leading terms of a tuple F of polynomials in I. In order to efficiently find such tuples, we provide two major new tools: an algorithm which reduces the check whether a given tuple F is Z-separating to an LP feasibility problem, and an isomorphism between the part of the Gröbner fan of I consisting of marked reduced Gröbner bases which contain a Z-separating tuple and the Gröbner fan of the intersection of I and K[X\Z]. We also indicate a possible generalization to tuples Z which consist of terms. All results are illustrated by explicit examples.

math.AC↗

arXiv2021

The Axiomatics of Free Group Rings

In [FGRS1,FGRS2] the relationship between the universal and elementary theory of a group ring $R[G]$ and the corresponding universal and elementary theory of the associated group $G$ and ring $R$ was examined. Here we assume that $R$ is a commutative ring with identity $1 \ne 0$. Of course, these are relative to an appropriate logical language $L_0,L_1,L_2$ for groups, rings and group rings respectively. Axiom systems for these were provided in [FGRS1]. In [FGRS1] it was proved that if $R[G]$ is elementarily equivalent to $S[H]$ with respect to $L_{2}$, then simultaneously the group $G$ is elementarily equivalent to the group $H$ with respect to $L_{0}$, and the ring $R$ is elementarily equivalent to the ring $S$ with respect to $L_{1}$. We then let $F$ be a rank $2$ free group and $\mathbb{Z}$ be the ring of integers. Examining the universal theory of the free group ring ${\mathbb Z}[F]$ the hazy conjecture was made that the universal sentences true in ${\mathbb Z}[F]$ are precisely the universal sentences true in $F$ modified appropriately for group ring theory and the converse that the universal sentences true in $F$ are the universal sentences true in ${\mathbb Z}[F]$ modified appropriately for group theory. In this paper we show this conjecture to be true in terms of axiom systems for ${\mathbb Z}[F]$.

math.GR↗

arXiv2021

Cotangent spaces and separating re-embeddings

Given an affine algebra $R=P/I$, where $P=K[x_1,\dots,x_n]$ is a polynomial ring over a field $K$ and $I$ is an ideal in $P$, we study re-embeddings of the affine scheme ${\rm Spec}(R)$, i.e., presentations $R \cong P'/I'$ such that $P'$ is a polynomial ring in fewer indeterminates. To find such re-embeddings, we use polynomials $f_i$ in the ideal $I$ which are coherently separating in the sense that they are of the form $f_i= z_i - g_i$ with an indeterminate $z_i$ which divides neither a term in the support of $g_i$ nor in the support of $f_j$ for $j\ne i$. The possible numbers of such sets of polynomials are shown to be governed by the Gröbner fan of $I$. The dimension of the cotangent space of $R$ at a $K$-linear maximal ideal is a lower bound for the embedding dimension, and if we find coherently separating polynomials corresponding to this bound, we know that we have determined the embedding dimension of $R$ and found an optimal re-embedding.

math.AC↗