TY - JOUR
T1 - A Lower Bound for the Determinantal Complexity of a Hypersurface
AU - Alper, Jarod
AU - Bogart, Tristram
AU - Velasco, Mauricio
N1 - Publisher Copyright:
© 2015, SFoCM.
PY - 2017/6/1
Y1 - 2017/6/1
N2 - We prove that the determinantal complexity of a hypersurface of degree d> 2 is bounded below by one more than the codimension of the singular locus, provided that this codimension is at least 5. As a result, we obtain that the determinantal complexity of the 3 × 3 permanent is 7. We also prove that for n> 3 , there is no nonsingular hypersurface in Pn of degree d that has an expression as a determinant of a d× d matrix of linear forms, while on the other hand for n≤ 3 , a general determinantal expression is nonsingular. Finally, we answer a question of Ressayre by showing that the determinantal complexity of the unique (singular) cubic surface containing a single line is 5.
AB - We prove that the determinantal complexity of a hypersurface of degree d> 2 is bounded below by one more than the codimension of the singular locus, provided that this codimension is at least 5. As a result, we obtain that the determinantal complexity of the 3 × 3 permanent is 7. We also prove that for n> 3 , there is no nonsingular hypersurface in Pn of degree d that has an expression as a determinant of a d× d matrix of linear forms, while on the other hand for n≤ 3 , a general determinantal expression is nonsingular. Finally, we answer a question of Ressayre by showing that the determinantal complexity of the unique (singular) cubic surface containing a single line is 5.
KW - Affine linear projections
KW - Cubic surfaces
KW - Determinantal complexity
KW - Permanents
UR - http://www.scopus.com/inward/record.url?scp=84947601790&partnerID=8YFLogxK
U2 - 10.1007/s10208-015-9300-x
DO - 10.1007/s10208-015-9300-x
M3 - Article
SN - 1615-3375
VL - 17
SP - 829
EP - 836
JO - Foundations of Computational Mathematics
JF - Foundations of Computational Mathematics
IS - 3
ER -