Indian Institute of Technology Palakkad&nbsp;(IIT Palakkad or IIT PKD) is a public autonomous&nbsp;engineering&nbsp;and&nbsp;research&nbsp;institute located in&nbsp;Palakkad,&nbsp;Kerala. It was&nbsp;one of the five new&nbsp;IITs&nbsp;proposed in the&nbsp;2014 Union budget of India. The campus was inaugurated on 3 August 2015. The institute has 94 faculty, 978 students, and 63 non-teaching staffs.

IT Palakkad currently offers four-year Bachelor of Technology (B.Tech) and two years MS programmes in 8 different engineering disciplines including&nbsp;Computer Science, Electrical, Mechanical, Civil among others.

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IIT Palakkad

Given a linear recurrence sequence (LRS), specified using the initial conditions and the recurrence relation, the Skolem problem asks if zero ever occurs in the infinite sequence generated by the LRS. Despite active research over last few decades, its decidability is known only for a few restricted subclasses, by either restricting the order of the LRS (upto 4) or by restricting the structure of the LRS (e.g., roots of its characteristic polynomial). In this paper, we identify a subclass of LRS of arbitrary order for which the Skolem problem is easy, namely LRS all of whose characteristic roots are (possibly complex) roots of real algebraic numbers, i.e., roots satisfying xd = r for r real algebraic. We show that for this subclass, the Skolem problem can be solved in NPRP. As a byproduct, we implicitly obtain effective bounds on the zero set of the LRS for this subclass. While prior works in this area often exploit deep results from algebraic and transcendental number theory to get such effective results, our techniques are primarily algorithmic and use linear algebra and Galois theory. We also complement our upper bounds with a NP lower bound for the Skolem problem via a new direct reduction from 3-CNF-SAT, matching the best known lower bounds. © S. Akshay, Nikhil Balaji, Aniket Murhekar, Rohith Varma, and Nikhil Vyas; licensed under Creative Commons License CC-BY

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Department of Mathematics

Leibniz International Proceedings in Informatics, LIPIcs

Near-optimal complexity bounds for fragments of the skolem problem

Let G be a parahoric group scheme over a complex projective curve X of genus greater than one. Let G denote the moduli stack of G-torsors on X. We prove several results concerning the Hitchin map on TG. We first show that the parahoric analogue of the global nilpotent cone is isotropic and use this to prove that G is "very good" in the sense of Beilinson-Drinfeld. We then prove that the parahoric Hitchin map is a Poisson map whose generic fibres are abelian varieties. Together, these results imply that the parahoric Hitchin map is a completely integrable system. © 2018 The Author(s).

Fulltext

International Mathematics Research Notices

Complete Integrability of the Parahoric Hitchin System

arXiv preprint arXiv:1607.

Global nilpotent cone is isotropic : parahoric torsors on curves

Let π: X → C be a relatively minimal non-isotrivial elliptic surface over the field of complex numbers, where g(C) ≥ 2. In this article, we demonstrate an equivalence between the category of semistable parabolic Higgs bundles on C, and the category of semistable Higgs bundles on X with vanishing second Chern class, and determinant a vertical divisor. © 2015. Published by Oxford University Press. All rights reserved.