arXiv subjects

Turgay Akyar

Publications and source records attributed to Turgay Akyar.

3 recordsLinked to original sources

arXiv2026

The 2-Adic Valuation of the Order of the All-Ones Class in the Sandpile Group of a Square

Place one grain at every nonsink vertex of the wired $n\times n$ square, and let $L(n)$ be the order of this operation in the sandpile group. Thus $L(n)$ is the least positive $q$ for which $q$ uniform grain layers form an integral combination of toppling moves. We prove that, for every $n\ge1$, \[ ν_2(L(n))= \begin{cases} 2,&n=1,\\ 1,&n\ge2\text{ even},\\ ν_2(n+1)+2,&n\ge3\text{ odd}. \end{cases} \] For even squares, this follows from the domino--sandpile results of Florescu, Morar, Perkinson, Salter, and Xu, completed by a short parity observation. For odd squares, a unimodular cyclic basis identifies the folded cokernel with a quotient by two shifted Chebyshev polynomials and sends the all-ones class to $1$. Its order is determined by the constant part of this polynomial ideal, not just by a determinant. Two normalized Euclidean remainders reduce to consecutive Fibonacci polynomials over $\mathbb F_2$, giving the exact valuation.

math.CO↗

arXiv2026

Introduction to non-Abelian Patchworking

The note introduces a novel concept of non-Abelian patchworking arising as real locus of non-Abelian complex-phase tropical hypersurfaces, the theory of which is now developed enough to allow the proposed spin-off. Although, non-Abelian Tropical Geometry makes sense for an arbitrary reductive complex group, the state of the art is that of full understanding of tropicalizations of surfaces within three dimensional groups $PGL_2(\mathbb{C})$ and $SL_2(\mathbb{C}),$ which are closely related via the two-fold covering. We stress our point, that this is an announcement of a framework, taking care of explaining explicitly the input, which is more geometric and less combinatorial than in the original Viro's method, to construct possible types of real algebraic surfaces in the real projective 3-space, and verify that it reproduces all the existing isotopy types of surfaces up to degree three. We obtain two general theorems concerning the topology of primitive PGL2 surfaces, observing in particular that they may have different Euler charteristic for a fixed degree greater than one, not necessarily equal to the signature of the corresponding complex surface, which would be the case for primitive combinatorial patchworking due to a result of Itenberg.

math.AG↗

arXiv2025

Special Divisors on Real Trigonal Curves

In this paper we examine the topology of Brill-Noether varieties associated to real trigonal curves. More precisely, we aim to count the connected components of the real locus of the varieties parametrizing linear systems of degree $d$ and dimension at least $r$. We do this count when the relations $m=g-d+r-1\leq d-2r-1$ are satisfied, where $m$ is the Maroni invariant and $g$ is the genus of the curve.

math.AG↗