arXiv subjects

Madhavan Venkatesh

Publications and source records attributed to Madhavan Venkatesh.

9 recordsLinked to original sources

arXiv2026

On Variable-Bounded Non-Linear Expansions of Presburger Arithmetic

We consider expansions of Presburger arithmetic with families of monadic polynomial predicates. (Examples of such predicates are the set of perfect squares, or the set of integers of the form $2n^3-5n+3$, etc.) Although the full attendant first-order theories are well known to be undecidable, very little is known when one restricts the number of variables. In the case of single-variable theories, we obtain positive results for the following two families of predicates: (i) for perfect fixed powers, decidability ofthe corresponding theory follows from the solvability of hyperellipticDiophantine equations; and (ii) for polynomials of degree at most three, we establish decidability by relying on the low genus of the resulting algebraic curves. Finally, we discuss limitations and hardness results (via encodings of longstanding open Diophantine problems) as soon as any of the above restrictions are lifted.

cs.LO↗

arXiv2026

Computing Picard Schemes

We present an algorithm to compute the torsion component $\mathrm{Pic}^τX$ of the Picard scheme of a smooth projective variety $X$ over a field $k$. Specifically, we describe $\mathrm{Pic}^τX$ as a closed subscheme of a projective space defined by explicit homogeneous polynomials. Furthermore, we compute the group scheme structure on $\mathrm{Pic}^τX$. As applications, we provide algorithms to compute various homological invariants. Among these, we compute the abelianization of the geometric étale fundamental group $π^{\mathrm{{e}t}}_1(X_{\bar{k}}, x)^{\mathrm{ab}}$. Moreover, we determine the Galois module structure of the first étale cohomology groups $H^1_{\mathrm{{e}t}}(X_{\bar{k}}, \mathbb{Z}/n\mathbb{Z})$ without requiring $n$ to be prime to the characteristic of $k$.

math.AG↗

arXiv2025

Counting points on surfaces in polynomial time

We present a randomised algorithm to compute the local zeta function of a fixed smooth, projective surface over $\mathbb{Q}$, at any large prime $p$ of good reduction. The runtime of our algorithm is polynomial in $\log p$, resolving a conjecture of Couveignes and Edixhoven.

math.NT↗

arXiv2025

Complexity of counting points on curves and the factor $P_1(T)$ of the zeta function of surfaces

This article concerns the computational complexity of a fundamental problem in number theory: counting points on curves and surfaces over finite fields. There is no subexponential-time algorithm known and it is unclear if it can be $\mathrm{NP}$-hard. Given a curve, we present the first efficient Arthur-Merlin protocol to certify its point-count, its Jacobian group structure, and its Hasse-Weil zeta function. We extend this result to a smooth projective surface to certify the factor $P_{1}(T)$, corresponding to the first Betti number, of the zeta function; by using the counting oracle. We give the first algorithm to compute $P_{1}(T)$ that is poly($\log q$)-time if the degree $D$ of the input surface is fixed; and in quantum poly($D\log q$)-time in general. Our technique in the curve case, is to sample hash functions using the Weil and Riemann-Roch bounds, to certify the group order of its Jacobian. For higher dimension varieties, we first reduce to the case of a surface, which is fibred as a Lefschetz pencil of hyperplane sections over $\mathbb{P}^{1}$. The formalism of vanishing cycles, and the inherent big monodromy, enable us to prove an effective version of Deligne's `theoreme du pgcd' using the hard-Lefschetz theorem and an equidistribution result due to Katz. These reduce our investigations to that of computing the zeta function of a curve, defined over a finite field extension $\mathbb{F}_{Q}/\mathbb{F}_{q}$ of poly-bounded degree. This explicitization of the theory yields the first nontrivial upper bounds on the computational complexity.

cs.CC↗

arXiv2025

Bornes de torsion et un théorème effectif du pgcd

We prove an effective, probabilistic version of Deligne's `théorème du pgcd' for a smooth, projective, geometrically integral (\textit{nice}) variety $X_{0}\subset \mathbb{P}^{N}$ over $\mathbb{F}_{q}$ of dimension $n$ and degree $D$, obtained via good reduction from a nice variety $\mathcal{X}_{0}$ over a number field $K$ at a prime $\mathfrak{p}\subset \mathcal{O}_{K}$. The main ingredients include bounding torsion in the Betti cohomology of $\mathcal{X}_{0}$, a mod -- $\ell$ big monodromy result and equidistribution of Frobenius in the representation associated to the sheaf of vanishing cycles modulo $\ell$.

math.AG↗

arXiv2022

A Construct of Dynamics, Space and Gravity from Loops

An attempt is made to construct space and obtain dynamics from Loop Algebras and their elements. We define three new products between loops namely 'Vertical Product', 'Horizontal Product' and 'Total Product'. As for the dynamics, we obtain corresponding "velocity" and "canonical momenta" from it. Also, we build a new "Energy Variable" that is dependent on the velocity and momentum alone. Then, we apply the loop constructs to General Relativity and arrive at the Einstein Field Equations, although presented in a different form. The key feature of this formalism is that the metric is not arbitarized as prevalent on the space beforehand but is rather induced by restricting the Killing Form to the Cartan Sub-algebra of the underlying Lie Algebra. Then we go on to show that "Dynamics is Structure" and that both do not depend on length or time.

gr-qc↗

arXiv2013

Loop Variable Representation of Classical Higher Dimensional Gravity and the Hilbert Space Grassmannian

In this paper, an attempt is made to represent 5+1 dimensional gravity (via ADM formalism) in terms of the loop constructions introduced by the author in a companion paper. The "momenta" and "velocity" from the earlier paper, which were proven to be cobordant loops in 6 D; are used as the new loop variables. In the process, the Hamiltonian, Diffeomorphism and Gauss constraints are written in polynomials of these loop variables. Other constraints such as the "Q" constraint and simplicity constraints arise due to greater degrees of freedom. We then undergo the Master Constraint treatment to resolve the constraints. Then, a pre-quantum version of the theory is examined; and the properties of the Grassmannian of the Hilbert Space are explored.

gr-qc↗

arXiv2013

An Algebraic Topological Construct of Classical Loop Gravity and the prospect of Higher Dimensions

In this paper, classical gravity is reformulated in terms of loops, via an algebraic topological approach. The main component is the loop group, whose elements consist of pairs of cobordant loops. A Chas-Sullivan product is described on the cobordism, and three other products, namely the 'vertical', 'horizontal' and 'total' products are re-introduced. (They have already been defined in an earlier paper by the author). A loop calculus is introduced on the space of loops, consisting of the loop variation functional,the loop derivative, Mandelstam derivative, and what the author wishes to call, the Gambini-Pullin contact functional. The loop derivative happens to be a generator of the group of loops, and the Gambini-Pullin functional is an infinitesimal generator of diffeomorphisms. A toy model of gravity is formulated in terms of the above, and it is proven that the total product provides for the Maurer-Cartan structure of the space. Further, new quantities labeled as 'momenta', 'velocity' and 'energy' are introduced in terms of the loop products and loop derivatives. The prospect of re-constructing general relativity in higher dimensions is explored, with the 'momenta' and 'velocity' being the basic loop variables. In the process, it is proven that the 'momenta' and 'velocity' behave as cobordant loops in higher dimensions.

gr-qc↗

arXiv2013

The Canonical Lagrangian Approach To Three-Space General Relativity

We study the action for the three-space formalism of General Relativity, better known as the BFÓ (Barbour--Foster--Ó Murchadha) action, which is a square-root BSW (Baierlein--Sharp--Wheeler) action. In particular, we explore the (pre)symplectic structure by pulling it back via a Legendre map to the tangent bundle of the configuration space of this action. With it we attain the canonical Lagrangian vector field which generates the gauge transformations (3-diffeomorphisms) and the true physical evolution of the system. This vector field encapsulates all the dynamics of the system. We also discuss briefly the observables and perennials for this theory. We then present a symplectic reduction of the constrained phase space.

gr-qc↗