Abstract
We prove a higher codimensional radical SylvesterGallai type theorem for quadratic polynomials, simultaneously generalizing [Hansen, 1965; Shpilka, 2020]. Hansen’s theorem is a highdimensional version of the classical SylvesterGallai theorem in which the incidence condition is given by highdimensional 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 highcodimensional ideal. Our main theorem is also a generalization of the quadratic SylvesterGallai Theorem of [Shpilka, 2020].
Our work is the first to prove a radical SylvesterGallai 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].
BibTeX  Entry
@InProceedings{garg_et_al:LIPIcs.CCC.2023.20,
author = {Garg, Abhibhav and Oliveira, Rafael and Peleg, Shir and Sengupta, Akash Kumar},
title = {{Radical SylvesterGallai Theorem for Tuples of Quadratics}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {20:120:30},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772822},
ISSN = {18688969},
year = {2023},
volume = {264},
editor = {TaShma, Amnon},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/18290},
URN = {urn:nbn:de:0030drops182903},
doi = {10.4230/LIPIcs.CCC.2023.20},
annote = {Keywords: SylvesterGallai theorem, arrangements of hypersurfaces, algebraic complexity, polynomial identity testing, algebraic geometry, commutative algebra}
}
Keywords: 

SylvesterGallai theorem, arrangements of hypersurfaces, algebraic complexity, polynomial identity testing, algebraic geometry, commutative algebra 
Collection: 

38th Computational Complexity Conference (CCC 2023) 
Issue Date: 

2023 
Date of publication: 

10.07.2023 