TY - GEN

T1 - A generalized Sylvester-Gallai type theorem for quadratic polynomials

AU - Peleg, Shir

AU - Shpilka, Amir

N1 - Publisher Copyright:
© Shir Peleg and Amir Shpilka; licensed under Creative Commons License CC-BY 35th Computational Complexity Conference (CCC 2020).

PY - 2020/7/1

Y1 - 2020/7/1

N2 - 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 Σ[3]ΠΣΠ[2] circuits. Specifically, we prove that if a finite set of irreducible quadratic polynomials Q satisfy that for every two polynomials Q1, Q2 ∈ Q there is a subset K ⊂ Q, such that Q1, Q2 ∈/ K and whenever Q1 and Q2 vanish then QQi∈K Qi vanishes, then the linear span of the polynomials in Q has dimension O(1). This extends the earlier result [33] that showed a similar conclusion when |K| = 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 [33] that studied the case when one quadratic polynomial is in the radical of two other quadratics.

AB - 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 Σ[3]ΠΣΠ[2] circuits. Specifically, we prove that if a finite set of irreducible quadratic polynomials Q satisfy that for every two polynomials Q1, Q2 ∈ Q there is a subset K ⊂ Q, such that Q1, Q2 ∈/ K and whenever Q1 and Q2 vanish then QQi∈K Qi vanishes, then the linear span of the polynomials in Q has dimension O(1). This extends the earlier result [33] that showed a similar conclusion when |K| = 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 [33] that studied the case when one quadratic polynomial is in the radical of two other quadratics.

KW - Algebraic computation

KW - Computational complexity

KW - Computational geometry

UR - http://www.scopus.com/inward/record.url?scp=85089390050&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.CCC.2020.8

DO - 10.4230/LIPIcs.CCC.2020.8

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AN - SCOPUS:85089390050

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 35th Computational Complexity Conference, CCC 2020

A2 - Saraf, Shubhangi

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 35th Computational Complexity Conference, CCC 2020

Y2 - 28 July 2020 through 31 July 2020

ER -