Related papers: On the Gr\"obner complexity of matrices

In 1980, White conjectured that the toric ideal of a matroid is generated by quadratic binomials corresponding to a symmetric exchange. In this paper, we compute Gr\"obner bases of toric ideals associated with matroids and show that, for…

In this paper, we study uniqueness properties of solutions to the generalized fourth-order Schr\"odinger equations in any dimension $d$ of the following forms, $$i \partial_t u + \sum_{j=1}^d \partial_{x_j}^{\, 4} u = V(t, x) u, \quad…

In this paper we give necessary and sufficient trace conditions for an n by n matrix over any commutative and associative ring with unity to be a sum of k-th powers of matrices over that ring, where n,k are integers greater equal 2. We…

A question of interest in Linear Algebra is whether all n x n complex matrices can be unitarily tridiagonalised. The answer for all n not equal to 4 (affirmative or negative) has been known for a while, whereas the case n=4 seems to have…

We present algebraic projective geometry definitions of 3D rotations so as to bridge a small gap between the applications and the definitions of 3D rotations in homogeneous matrix form. A general homogeneous matrix formulation to 3D…

We study intersection matrix algebras im(A^d) that arise from affinizing a Cartan matrix A of type B_r with d arbitrary long roots in the root system $\Delta_{B_r}$, where $r \geq 3$. We show that im(A^d) is isomorphic to the universal…

Let $M$ be an $n\times n$ matrix of homogeneous linear forms over a field $\Bbbk$. If the ideal $\mathcal{I}_{n-2}(M)$ generated by minors of size $n-1$ is Cohen-Macaulay, then the Gulliksen-Neg{\aa}rd complex is a free resolution of…

The Euler quotient modulo an odd-prime power $p^r~(r>1)$ can be uniquely decomposed as a $p$-adic number of the form $$ \frac{u^{(p-1)p^{r-1}} -1}{p^r}\equiv a_0(u)+a_1(u)p+\ldots+a_{r-1}(u)p^{r-1} \pmod {p^r},~ \gcd(u,p)=1, $$ where $0\le…

It was conjectured by Flynn that there exists a constant $\kappa$ such that, for any integer $g \ge 2$, any $m \le \kappa g$, there exists a hyperelliptic curve of genus $g$ over $\mathbb Q$ with a rational $m$-torsion point on its…

For each integer partition $\mathbf{q}$ with $d$ parts, we denote by $\Delta_{(1,\mathbf{q})}$ the lattice simplex obtained as the convex hull in $\mathbb{R}^d$ of the standard basis vectors along with the vector $-\mathbf{q}$. For…

Over the past decade, the Gr\"obner basis theory and automatic solver generation have lead to a large number of solutions to geometric vision problems. In practically all cases, the derived solvers apply a fixed elimination template to…

We present a formalization of Gr\"obner basis theory in Lean 4, built on top of Mathlib's infrastructure for multivariate polynomials and monomial orders. Our development covers the core foundations of Gr\"obner basis theory, including…

Let $A$ be an $N\times N$ irreducible matrix with entries in $\{0,1\}$. We present an easy way to find an $(N+3)\times (N+3)$ irreducible matrix $\bar{A}$ with entries in $\{0,1\}$ such that their Cuntz--Krieger algebras ${\mathcal{O}}_A$…

In this note, we extend modular techniques for computing Gr\"obner bases from the commutative setting to the vast class of noncommutative $G$-algebras. As in the commutative case, an effective verification test is only known to us in the…

We show that for any integer $n\geq2$ there is a smooth complex projective variety $X$ of dimension $5$ whose third Griffiths group $\text{Griff}^{3}(X)$ contains infinitely many torsion elements of order $n$. This generalises a recent…

Formal verification techniques based on computer algebra have proven highly effective for circuit verification. The circuit, given as an and-inverter graph, is encoded as a set of polynomials that automatically generates a Gr\"obner basis…

We construct the Hasse diagrams $G_2$ and $G_3$ for the closure ordering on the sets of congruence classes of $2\times 2$ and $3\times 3$ complex matrices. In other words, we construct two directed graphs whose vertices are $2\times 2$ or,…

We study the computational complexity of the general network satisfaction problem for a finite relation algebra $A$ with a normal representation $B$. If $B$ contains a non-trivial equivalence relation with a finite number of equivalence…

By conformal welding, there is a pair of univalent functions $(f,g)$ associated to every point of the complex K\"ahler manifold $\Mob(S^1)\bk\Diff_+(S^1)$. For every integer $n\geq 1$, we generalize the definition of Faber polynomials to…

We describe the equations and Gr\"obner bases of some degenerate K3 surfaces associated to rational normal scrolls. These K3 surfaces are members of a class of interesting singular projective varieties we call correspondence scrolls. The…