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Register a new userGeometric Complexity Theory V: Efficient algorithms for Noether Normalization
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Ketan Mulmuley D
Publication date
2012
fields
Informatics Engineering
and research's language is
English
Authors
Ketan D. Mulmuley
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We study a basic algorithmic problem in algebraic geometry, which we call NNL, of constructing a normalizing map as per Noethers Normalization Lemma. For general explicit varieties, as formally defined in this paper, we give a randomized polynomial-time Monte Carlo algorithm for this problem. For some interesting cases of explicit varieties, we give deterministic quasi-polynomial time algorithms. These may be contrasted with the standard EXPSPACE-algorithms for these problems in computational algebraic geometry. In particular, we show that: (1) The categorical quotient for any finite dimensional representation $V$ of $SL_m$, with constant $m$, is explicit in characteristic zero. (2) NNL for this categorical quotient can be solved deterministically in time quasi-polynomial in the dimension of $V$. (3) The categorical quotient of the space of $r$-tuples of $m times m$ matrices by the simultaneous conjugation action of $SL_m$ is explicit in any characteristic. (4) NNL for this categorical quotient can be solved deterministically in time quasi-polynomial in $m$ and $r$ in any characteristic $p$ not in $[2, m/2]$. (5) NNL for every explicit variety in zero or large enough characteristic can be solved deterministically in quasi-polynomial time, assuming the hardness hypothesis for the permanent in geometric complexity theory. The last result leads to a geometric complexity theory approach to put NNL for every explicit variety in P.
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These are lectures notes for the introductory graduate courses on geometric complexity theory (GCT) in the computer science department, the university of Chicago. Part I consists of the lecture notes for the course given by the first author in the spring quarter, 2007. It gives introduction to the basic structure of GCT. Part II consists of the lecture notes for the course given by the second author in the spring quarter, 2003. It gives introduction to invariant theory with a view towards GCT. No background in algebraic geometry or representation theory is assumed. These lecture notes in conjunction with the article cite{GCTflip1}, which describes in detail the basic plan of GCT based on the principle called the flip, should provide a high level picture of GCT assuming familiarity with only basic notions of algebra, such as groups, rings, fields etc.
This article has been withdrawn because it has been merged with the earlier article GCT3 (arXiv: CS/0501076 [cs.CC]) in the series. The merged article is now available as: Geometric Complexity Theory III: on deciding nonvanishing of a Littlewood-Richardson Coefficient, Journal of Algebraic Combinatorics, vol. 36, issue 1, 2012, pp. 103-110. (Authors: Ketan Mulmuley, Hari Narayanan and Milind Sohoni) The new article in this GCT5 slot in the series is: Geometric Complexity Theory V: Equivalence between blackbox derandomization of polynomial identity testing and derandomization of Noethers Normalization Lemma, in the Proceedings of FOCS 2012 (abstract), arXiv:1209.5993 [cs.CC] (full version) (Author: Ketan Mulmuley)
The problem of graph Reachability is to decide whether there is a path from one vertex to another in a given graph. In this paper, we study the Reachability problem on three distinct graph families - intersection graphs of Jordan regions, unit contact disk graphs (penny graphs), and chordal graphs. For each of these graph families, we present space-efficient algorithms for the Reachability problem. For intersection graphs of Jordan regions, we show how to obtain a good vertex separator in a space-efficient manner and use it to solve the Reachability in polynomial time and $O(m^{1/2}log n)$ space, where $n$ is the number of Jordan regions, and $m$ is the total number of crossings among the regions. We use a similar approach for chordal graphs and obtain a polynomial-time and $O(m^{1/2}log n)$ space algorithm, where $n$ and $m$ are the number of vertices and edges, respectively. However, we use a more involved technique for unit contact disk graphs (penny graphs) and obtain a better algorithm. We show that for every $epsilon> 0$, there exists a polynomial-time algorithm that can solve Reachability in an $n$ vertex directed penny graph, using $O(n^{1/4+epsilon})$ space. We note that the method used to solve penny graphs does not extend naturally to the class of geometric intersection graphs that include arbitrary size cliques.
This article describes a {em nonstandard} quantum group that may be used to derive a positive formula for the plethysm problem, just as the standard (Drinfeld-Jimbo) quantum group can be used to derive the positive Littlewood-Richardson rule for arbitrary complex semisimple Lie groups. The sequel cite{GCT8} gives conjecturally correct algorithms to construct canonical bases of the coordinate rings of these nonstandard quantum groups and canonical bases of the dually paired nonstandard deformations of the symmetric group algebra. A positive $#P$-formula for the plethysm constant follows from the conjectural properties of these canonical bases and the duality and reciprocity conjectures herein.
Geometric complexity theory (GCT) is an approach to the P vs. NP and related problems. This article gives its complexity theoretic overview without assuming any background in algebraic geometry or representation theory.
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