TY - GEN

T1 - Robust 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 4.0

PY - 2022/6/1

Y1 - 2022/6/1

N2 - In this work we extend the robust version of the Sylvester-Gallai theorem, obtained by Barak, Dvir, Wigderson and Yehudayoff, and by Dvir, Saraf and Wigderson, to the case of quadratic polynomials. Specifically, we prove that if Q ? C[x1...., xn] is a finite set, |Q| = m, of irreducible quadratic polynomials that satisfy the following condition There is d > 0 such that for every Q ? Q there are at least dm polynomials P ? Q such that whenever Q and P vanish then so does a third polynomial in Q \ (Q, P). then dim(span(Q)) = Poly(1/d). The work of Barak et al. and Dvir et al. studied the case of linear polynomials and proved an upper bound of O(1/d) on the dimension (in the first work an upper bound of O(1/d2) was given, which was improved to O(1/d) in the second work).

AB - In this work we extend the robust version of the Sylvester-Gallai theorem, obtained by Barak, Dvir, Wigderson and Yehudayoff, and by Dvir, Saraf and Wigderson, to the case of quadratic polynomials. Specifically, we prove that if Q ? C[x1...., xn] is a finite set, |Q| = m, of irreducible quadratic polynomials that satisfy the following condition There is d > 0 such that for every Q ? Q there are at least dm polynomials P ? Q such that whenever Q and P vanish then so does a third polynomial in Q \ (Q, P). then dim(span(Q)) = Poly(1/d). The work of Barak et al. and Dvir et al. studied the case of linear polynomials and proved an upper bound of O(1/d) on the dimension (in the first work an upper bound of O(1/d2) was given, which was improved to O(1/d) in the second work).

KW - Algebraic computation

KW - Sylvester-Gallai theorem

KW - quadratic polynomials

UR - http://www.scopus.com/inward/record.url?scp=85134332186&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.SoCG.2022.43

DO - 10.4230/LIPIcs.SoCG.2022.43

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AN - SCOPUS:85134332186

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 38th International Symposium on Computational Geometry, SoCG 2022

A2 - Goaoc, Xavier

A2 - Kerber, Michael

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

Y2 - 7 June 2022 through 10 June 2022

ER -