TY - JOUR

T1 - A Lower Bound on Determinantal Complexity

AU - Kumar, Mrinal

AU - Volk, Ben Lee

N1 - Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

PY - 2022/12

Y1 - 2022/12

N2 - The determinantal complexity of a polynomial P∈ F[x1, … , xn] over a field F is the dimension of the smallest matrix M whose entries are affine functions in F[x1, … , xn] such that P= Det (M). We prove that the determinantal complexity of the polynomial ∑i=1nxin is at least 1.5 n- 3.For every n-variate polynomial of degree d, the determinantal complexity is trivially at least d, and it is a long-standing open problem to prove a lower bound which is super linear in max { n, d}. Our result is the first lower bound for any explicit polynomial which is bigger by a constant factor than max { n, d} , and improves upon the prior best bound of n+ 1 , proved by Alper et al. (2017) for the same polynomial.

AB - The determinantal complexity of a polynomial P∈ F[x1, … , xn] over a field F is the dimension of the smallest matrix M whose entries are affine functions in F[x1, … , xn] such that P= Det (M). We prove that the determinantal complexity of the polynomial ∑i=1nxin is at least 1.5 n- 3.For every n-variate polynomial of degree d, the determinantal complexity is trivially at least d, and it is a long-standing open problem to prove a lower bound which is super linear in max { n, d}. Our result is the first lower bound for any explicit polynomial which is bigger by a constant factor than max { n, d} , and improves upon the prior best bound of n+ 1 , proved by Alper et al. (2017) for the same polynomial.

KW - 68Q06

KW - 68Q15

KW - 68Q17

KW - Algebraic Circuits

KW - Algebraic Complexity Theory

KW - Determinantal Complexity

KW - Lower Bounds

UR - http://www.scopus.com/inward/record.url?scp=85138225413&partnerID=8YFLogxK

U2 - 10.1007/s00037-022-00228-3

DO - 10.1007/s00037-022-00228-3

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AN - SCOPUS:85138225413

SN - 1016-3328

VL - 31

JO - Computational Complexity

JF - Computational Complexity

IS - 2

M1 - 12

ER -