TY - JOUR

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

AU - Peleg, Shir

AU - Shpilka, Amir

N1 - Publisher Copyright:
© 2022 The Author(s).

PY - 2022/12/15

Y1 - 2022/12/15

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 satisfies that for every two polynomials Q1, Q2 ∈ Q there is a subset κ ⊂ Q such that Q1, Q2 ∈ κ and whenever Q1 and Q2 vanish, then Πi∈κ vanishes, then the linear span of the polynomials in Q has dimension Q(1). This extends the earlier result [21] that holds for the case |κ| = 1.

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 satisfies that for every two polynomials Q1, Q2 ∈ Q there is a subset κ ⊂ Q such that Q1, Q2 ∈ κ and whenever Q1 and Q2 vanish, then Πi∈κ vanishes, then the linear span of the polynomials in Q has dimension Q(1). This extends the earlier result [21] that holds for the case |κ| = 1.

UR - http://www.scopus.com/inward/record.url?scp=85144341068&partnerID=8YFLogxK

U2 - 10.1017/fms.2022.100

DO - 10.1017/fms.2022.100

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AN - SCOPUS:85144341068

SN - 2050-5094

VL - 10

JO - Forum of Mathematics, Sigma

JF - Forum of Mathematics, Sigma

M1 - e112

ER -