arXiv subjects

Tanush Shaska

Publications and source records attributed to Tanush Shaska.

11 recordsLinked to original sources

arXiv2026

Arithmetic Sparsity and Obstructions in Weighted Projective Spaces

Let $\mathbb{WP}^n_{\mathbf{q}}$ be a weighted projective space with weights $\mathbf{q} = (q_0, \dots, q_n)$, $q = \operatorname{lcm}(q_i)$, and let $ϕ\colon \mathbb{WP}^n_{\mathbf{q}} \to \mathbb{P}^n$, $[x_i] \mapsto [x_i^{q/q_i}]$, be the Veronese morphism. A point of $\mathbb{P}^n(\mathbb{Q})$ is the image of a rational point of $\mathbb{WP}^n_{\mathbf{q}}$ only if its valuation vector at every prime satisfies a Kummer congruence. We count the rational points of $\mathbb{WP}^n_{\mathbf{q}}$ of bounded weighted height $\mathfrak{h} = H(ϕ(\,\cdot\,))^{1/q}$ and prove that, on the locus where all coordinates are nonzero, $$ Z^{\circ}_{\mathfrak{h}}\big( \mathbb{WP}^n_{\mathbf{q}}(\mathbb{Q}), X \big) = X^{q\,a(\mathbf{q})} P_{\mathbf{q}}(\log X) + O\big( X^{q\,a(\mathbf{q}) - θ} \big), \qquad θ> 0, $$ with $P_{\mathbf{q}}$ of exact degree $β(\mathbf{q})$, where $a(\mathbf{q})$ and $β(\mathbf{q})$ are the value and the dimension of the optimal face of a linear program determined by the Kummer congruences. The exponent satisfies $Q \leq q\,a(\mathbf{q}) \leq q(n+1)$, $Q = \sum q_i$, with equality on the right if and only if the exponents $q/q_i$ are pairwise coprime; the difference $q(n+1) - q\,a(\mathbf{q})$ measures the sparsity of the rational points of $\mathbb{WP}^n_{\mathbf{q}}$ relative to those of its Veronese image. The leading constant is evaluated when the dual optimum is diagonal and for the weights $(2,2,3,3)$. The full counting function follows by stratification, and we formulate the conjecture over an arbitrary number field.

math.NT↗

arXiv2026

Hulls, linear equivalence, and weighted superelliptic codes

The containment of the code of the meet $G\wedge A$ in the hull and the identity $G\vee A-D=K-G\wedge A$ exchanging meet and join are known; imposing that $G\wedge A$ be principal constructs algebraic geometry codes with one-dimensional hull. We turn that construction into a measurement. For arbitrary divisors $G$ and $A$ we compute $C_L(D,G)\cap C_L(D,A)$ exactly: it is the code of the meet together with an excess $\varepsilon(G,A)$, canonically their quotient and a subquotient of $H^1(\mathcal O(G\wedge A))$. So $\varepsilon$ vanishes exactly when the meet is non-special; otherwise it certifies that $K-G\wedge A$ is linearly equivalent to an effective divisor, at degree zero the vanishing of a single class in the Picard group: the hull detects a linear equivalence rather than being built from one. For superelliptic curves $y^n=f(x)$ both sides can be computed: their weighted plane models in $\mathbb P^2_{(1,n/c,d/c)}$, $c=\gcd(n,d)$, identify codes $C_s$ of weighted forms of degree $s$ with those of $sD_\infty$ and turn hulls into lattice counts. The range on which $\varepsilon$ is blind is an explicit interval of degrees, where $\dim\operatorname{Hull}(C_s)=cμ(s)-nδ+1-g_X$, $μ(s)=\min\{s,M-s\}$, depends only on its affine-point count. Outside it the meet and join are invariant under $s\mapsto M-s$ while $\varepsilon$ is not, so every asymmetry of the hull profile is excess and the threshold in $s$ refines the divisor class: two totally split curves of genus two, over $\mathbb F_7$ and over $\mathbb F_{11}$, present the same class at the same pair of degrees and are separated by the profile alone. If $0\leq°(G\wedge A)\leq2g_X-2$ the hull is at most $g_X+1$, so it is large only where it is blind, and over a prime field, under an explicit inequality on $(n,d,q)$, its maximum over the family is $\ell(\lfloor M/2\rfloor D_\infty)$, attained exactly on the totally split locus.

math.AG↗

arXiv2026

Exact Quotients of Fresnel-Kummer Surfaces and Certified Biaxial Refraction

The Fresnel wave surface governs the propagation of light in a transparent biaxial crystal. It is a special Kummer quartic, and we identify it exactly. Over the complex numbers the wave surface of a crystal is the Kummer surface of the Jacobian of an explicit genus-two curve branched at the signed square roots of the three principal permittivities. This Jacobian is isogenous, by an isogeny with kernel of order four, to a product of two elliptic curves. One elliptic curve carries the three permittivities, and the other carries the optic-axis angle. The physical family is Zariski dense in the locus of genus-two curves with an extra involution, and its automorphism strata are explicit. The identification instantiates a task-aware quotient, which identifies parameters that differ by a nuisance transformation and carries invariant coordinates and explicit strata. For biaxial crystals, two ratios of the permittivities form a complete invariant of the wave surface up to rotation and rescaling, and the four real nodes are given in closed form. At an interface the candidate transmitted waves are the roots of a quartic of exact degree four. Its real-root count, root order, and repeated-root events are decided by exact algebraic predicates, and along the generic single-node encounters of the paper its discriminant vanishes to second order. Floating-point solvers drop forward transmitted modes near the optic axes, and the certified solver does not. On exact equivalence classes, learned models on quotient coordinates are invariant and more accurate than models on raw tensors, while learned root-count predicates fail near the optic axes.

cs.CV↗

arXiv2026

Orbits and Fields of Definition for Graded Keller Maps

We classify graded Keller maps by the signature of their weight vectors and address which hyperbolic weights support counterexamples and how the nonempty loci are stratified. We construct two families. Cyclic cones populate $K(3,(1,-1,-1))$ at every composite generic fibre degree $N\geq 6$, including a six-sheeted member over $\mathbb{Q}(\sqrt{-15})$ and a 33-sheeted member over $\mathbb{Q}$. A diagonal family gives $K(3,(1,-p,-p))\neq\varnothing$ for every $p\geq2$, with $\det JF_p=-p$ and generic fibre degree $2p-1$. The geometric monodromy of every member of both families is alternating or symmetric of full degree; hence the solvable case occurs only in generic degree three. At generic degree twelve, normalized quartic cyclic seeds form a smooth plane cubic over $\mathbb{Q}$, whose points yield pairwise distinct graded-equivalence orbits. Thus the relevant orbit space contains a nonconstant algebraic family, establishing genuine moduli rather than isolated examples."

math.AG↗

arXiv2026

Orbit counts and twist zeta functions of weighted projective stacks over finite fields

Let ${\mathbb F}_q$ be a finite field and $\mathbf{w}=(w_0,\dots,w_n)$ a vector of positive integer weights. Several finite-field counts attached to the weighted projective space $\mathbb{P}^n_\mathbf{w}$ are easily conflated: the coarse rational-point count and the stacky mass are both weight-independent, whereas the number $A_\mathbf{w}(q)$ of ${\mathbb F}_q^\times$-orbits on nonzero ${\mathbb F}_q$-representatives for the weighted action depends on the weights. We prove the closed formula \[ A_\mathbf{w}(q)=\sum_{\emptyset\ne S\subseteq\{0,\dots,n\}}(q-1)^{|S|-1}\gcd(k_S,q-1), \qquad k_S=\gcd\{w_i:i\in S\}, \] and identify $A_\mathbf{w}(q)$ intrinsically as the number of ${\mathbb F}_q$-isomorphism classes of the weighted projective stack $\mathcal{P}_\mathbf{w}=[({\mathbb A}^{n+1}\setminus\{0\})/\mathbb{G}_m]$ -- equivalently, the number of ${\mathbb F}_q$-twists lying over the coarse points -- the discrepancy from the coarse count being governed by the Kummer groups ${\mathbb F}_q^\times/({\mathbb F}_q^\times)^{k_S}$. We read the behaviour of $A_\mathbf{w}$ under reduction of the weight vector through the ${\mathbb F}_q$-cohomology of the associated $\boldsymbolμ_d$-gerbe, and prove that the twist zeta function $Z_{\mathrm{tw}}(\mathcal{P}_\mathbf{w},t)=\exp\bigl(\sum_{r\ge1}A_\mathbf{w}(q^r)t^r/r\bigr)$ is rational with multiplicity spectrum independent of $q$, admitting a single global functional equation precisely when the weights share a common prime-to-$p$ part -- for reduced $\mathbf{w}$, precisely when the twist theory is trivial. For weighted diagonal hypersurfaces and same-degree pairs in the split regime, we compute the twist zeta function of the substack explicitly, with reciprocal roots given by Gauss sums and, for intersections, by Frobenius eigenvalues of superelliptic curves.

math.AG↗

arXiv2021

Superelliptic curves with minimal weighted moduli height

For a superelliptic curve $\mathcal X$, defined over $\mathbb Q$, let $\mathfrak p$ denote the corresponding moduli point in the weighted moduli space. We describe a method how to determine a minimal integral model of $\mathcal X$ such that: i) the corresponding moduli point $\mathfrak p$ has minimal weighted height, ii) the equation of the curve has minimal coefficients. Part i) is accomplished by reduction of the moduli point which is equivalent with obtaining a representation of the moduli point $\mathfrak p$ with minimal weighted height, as defined in [5], and part ii) by the classical reduction of the binary forms.

math.NT↗

arXiv2016

Mësimdhënia e matematikës nëpërmjet problemeve klasike

In this paper we discuss how teaching of mathematics for middle school and high school students can be improved dramatically when motivation of concepts and ideas is done through the classical problems and the history of mathematics. This method improves intuition of students, awakens their curiosity, avoids memorizing useless formulas, and put concepts in a historical prospective. To illustrate we show how diagonalizing quadratic forms, elliptic integrals, discriminants of high degree polynomials, and geometric constructions can be introduced successfully in high school level.

math.HO↗

arXiv2012

Invariants of Binary Forms

Basic invariants of binary forms over $\mathbb C$ up to degree 6 (and lower degrees) were constructed by Clebsch and Bolza in the 19-th century using complicated symbolic calculations. Igusa extended this to algebraically closed fields of any characteristic using difficult techniques of algebraic geometry. In this paper a simple proof is supplied that works in characteristic $p > 5$ and uses some concepts of invariant theory developed by Hilbert (in characteristic 0) and Mumford, Haboush et al. in positive characteristic. Further the analogue for pairs of binary cubics is also treated.

math.AG↗

arXiv2006

Galois groups of prime degree polynomials with nonreal roots

In the process of computing the Galois group of a prime degree polynomial $f(x)$ over $\mathbb Q$ we suggest a preliminary checking for the existence of non-real roots. If $f(x)$ has non-real roots, then combining a 1871 result of Jordan and the classification of transitive groups of prime degree which follows from CFSG we get that the Galois group of $f(x)$ contains $A_p$ or is one of a short list. Let $f(x)\in \mathbb Q [x] $ be an irreducible polynomial of prime degree $p \geq 5$ and $r=2s$ be the number of non-real roots of $f(x)$. We show that if $s$ satisfies $s (s \log s + 2 \log s + 3) \leq p $ then $Gal (f)= A_p, S_p$.

math.GR↗

arXiv2006

Computational algebra and algebraic curves

The development of computational techniques in the last decade has made possible to attack some classical problems of algebraic geometry. In this survey, we briefly describe some open problems related to algebraic curves which can be approached from a computational viewpoint.

math.AC↗

arXiv2006

Computational Aspects of Hyperelliptic Curves

We introduce a new approach of computing the automorphism group and the field of moduli of points $\p=[C]$ in the moduli space of hyperelliptic curves $\H_g$. Further, we show that for every moduli point $\p \in \H_g(L)$ such that the reduced automorphism group of $\p$ has at least two involutions, there exists a representative $C$ of the isomorphism class $\p$ which is defined over $L$.

math.AG↗