eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-07-10
20:1
20:30
10.4230/LIPIcs.CCC.2023.20
article
Radical Sylvester-Gallai Theorem for Tuples of Quadratics
Garg, Abhibhav
1
https://orcid.org/0000-0001-9084-7499
Oliveira, Rafael
1
https://orcid.org/0000-0001-8917-8689
Peleg, Shir
2
Sengupta, Akash Kumar
3
Cheriton School of Computer Science, University of Waterloo, Canada
Blavatnik School of Computer Science, Tel Aviv University, Israel
Department of Mathematics, Columbia University, New York, NY, USA
We prove a higher codimensional radical Sylvester-Gallai type theorem for quadratic polynomials, simultaneously generalizing [Hansen, 1965; Shpilka, 2020]. Hansen’s theorem is a high-dimensional version of the classical Sylvester-Gallai theorem in which the incidence condition is given by high-dimensional flats instead of lines. We generalize Hansen’s theorem to the setting of quadratic forms in a polynomial ring, where the incidence condition is given by radical membership in a high-codimensional ideal. Our main theorem is also a generalization of the quadratic Sylvester-Gallai Theorem of [Shpilka, 2020].
Our work is the first to prove a radical Sylvester-Gallai type theorem for arbitrary codimension k ≥ 2, whereas previous works [Shpilka, 2020; Shir Peleg and Amir Shpilka, 2020; Shir Peleg and Amir Shpilka, 2021; Garg et al., 2022] considered the case of codimension 2 ideals. Our techniques combine algebraic geometric and combinatorial arguments. A key ingredient is a structural result for ideals generated by a constant number of quadratics, showing that such ideals must be radical whenever the quadratic forms are far apart. Using the wide algebras defined in [Garg et al., 2022], combined with results about integral ring extensions and dimension theory, we develop new techniques for studying such ideals generated by quadratic forms. One advantage of our approach is that it does not need the finer classification theorems for codimension 2 complete intersection of quadratics proved in [Shpilka, 2020; Garg et al., 2022].
https://drops.dagstuhl.de/storage/00lipics/lipics-vol264-ccc2023/LIPIcs.CCC.2023.20/LIPIcs.CCC.2023.20.pdf
Sylvester-Gallai theorem
arrangements of hypersurfaces
algebraic complexity
polynomial identity testing
algebraic geometry
commutative algebra