8 research outputs found
A Generalized Sylvester-Gallai Type Theorem for Quadratic Polynomials
In this work we prove a version of the Sylvester-Gallai theorem for quadratic polynomials that takes us one step closer to obtaining a deterministic polynomial time algorithm for testing zeroness of ?^{[3]}???^{[2]} circuits. Specifically, we prove that if a finite set of irreducible quadratic polynomials ? satisfy that for every two polynomials Q?,Q? ? ? there is a subset ? ? ?, such that Q?,Q? ? ? and whenever Q? and Q? vanish then ?_{Q_i??} Q_i vanishes, then the linear span of the polynomials in ? has dimension O(1). This extends the earlier result [Amir Shpilka, 2019] that showed a similar conclusion when |?| = 1.
An important technical step in our proof is a theorem classifying all the possible cases in which a product of quadratic polynomials can vanish when two other quadratic polynomials vanish. I.e., when the product is in the radical of the ideal generated by the two quadratics. This step extends a result from [Amir Shpilka, 2019] that studied the case when one quadratic polynomial is in the radical of two other quadratics
Strong Algebras and Radical Sylvester-Gallai Configurations
In this paper, we prove the following non-linear generalization of the
classical Sylvester-Gallai theorem. Let be an algebraically closed
field of characteristic , and be a set of irreducible homogeneous polynomials of
degree at most such that is not a scalar multiple of for . Suppose that for any two distinct , there is such that . We prove that such radical SG
configurations must be low dimensional. More precisely, we show that there
exists a function , independent of
and , such that any such configuration must
satisfy
Our result confirms a conjecture of Gupta [Gup14, Conjecture 2] and
generalizes the quadratic and cubic Sylvester-Gallai theorems of [S20,OS22].
Our result takes us one step closer towards the first deterministic polynomial
time algorithm for the Polynomial Identity Testing (PIT) problem for depth-4
circuits of bounded top and bottom fanins. Our result, when combined with the
Stillman uniformity type results of [AH20a,DLL19,ESS21], yields uniform bounds
for several algebraic invariants such as projective dimension, Betti numbers
and Castelnuovo-Mumford regularity of ideals generated by radical SG
configurations.Comment: 62 pages. Comments are welcome
Independence in Algebraic Complexity Theory
This thesis examines the concepts of linear and algebraic independence in algebraic complexity theory. Arithmetic circuits, computing multivariate polynomials over a field, form the framework of our complexity considerations. We are concerned with polynomial identity testing (PIT), the problem of deciding whether a given arithmetic circuit computes the zero polynomial. There are efficient randomized algorithms known for this problem, but as yet deterministic polynomial-time algorithms could be found only for restricted circuit classes. We are especially interested in blackbox algorithms, which do not inspect the given circuit, but solely evaluate it at some points. Known approaches to the PIT problem are based on the notions of linear independence and rank of vector subspaces of the polynomial ring. We generalize those methods to algebraic independence and transcendence degree of subalgebras of the polynomial ring. Thereby, we obtain efficient blackbox PIT algorithms for new circuit classes. The Jacobian criterion constitutes an efficient characterization for algebraic independence of polynomials. However, this criterion is valid only in characteristic zero. We deduce a novel Jacobian-like criterion for algebraic independence of polynomials over finite fields. We apply it to obtain another blackbox PIT algorithm and to improve the complexity of testing the algebraic independence of arithmetic circuits over finite fields.Die vorliegende Arbeit untersucht die Konzepte der linearen und algebraischen UnabhĂ€ngigkeit innerhalb der algebraischen KomplexitĂ€tstheorie. Arithmetische Schaltkreise, die multivariate Polynome ĂŒber einem Körper berechnen, bilden die Grundlage unserer KomplexitĂ€tsbetrachtungen. Wir befassen uns mit dem polynomial identity testing (PIT) Problem, bei dem entschieden werden soll ob ein gegebener Schaltkreis das Nullpolynom berechnet. FĂŒr dieses Problem sind effiziente randomisierte Algorithmen bekannt, aber deterministische Polynomialzeitalgorithmen konnten bisher nur fĂŒr eingeschrĂ€nkte Klassen von Schaltkreisen angegeben werden. Besonders von Interesse sind Blackbox-Algorithmen, welche den gegebenen Schaltkreis nicht inspizieren, sondern lediglich an Punkten auswerten. Bekannte AnsĂ€tze fĂŒr das PIT Problem basieren auf den Begriffen der linearen UnabhĂ€ngigkeit und des Rangs von UntervektorrĂ€umen des Polynomrings. Wir ĂŒbertragen diese Methoden auf algebraische UnabhĂ€ngigkeit und den Transzendenzgrad von Unteralgebren des Polynomrings. Dadurch erhalten wir effiziente Blackbox-PIT-Algorithmen fĂŒr neue Klassen von Schaltkreisen. Eine effiziente Charakterisierung der algebraischen UnabhĂ€ngigkeit von Polynomen ist durch das Jacobi-Kriterium gegeben. Dieses Kriterium ist jedoch nur in Charakteristik Null gĂŒltig. Wir leiten ein neues Jacobi-artiges Kriterium fĂŒr die algebraische UnabhĂ€ngigkeit von Polynomen ĂŒber endlichen Körpern her. Dieses liefert einen weiteren Blackbox-PIT-Algorithmus und verbessert die KomplexitĂ€t des Problems arithmetische Schaltkreise ĂŒber endlichen Körpern auf algebraische UnabhĂ€ngigkeit zu testen