Cited By
View all- Chillara SMukhopadhyay P(2019)Depth-4 Lower Bounds, Determinantal Complexity: A Unified ApproachComputational Complexity10.1007/s00037-019-00185-428:4(545-572)Online publication date: 1-Dec-2019
Lower Bounds for the Determinantal Complexity of Explicit Low Degree Polynomials
Pages 343 - 354
Abstract
The determinantal complexity of a polynomial f(x 1,x 2,…,x n ) is the minimum m such that f=det m (L(x 1,x 2,…,x n )), where L(x 1,x 2,…,x n ) is a matrix whose entries are affine forms in the x i s over some field $\mbox {$\mathbb {F}$}$.
Asymptotically tight lower bounds are proven for the determinantal complexity of the elementary symmetric polynomial $S^{d}_{n}$ of degree d in n variables, 2d-fold iterated matrix multiplication of the form 〈u|X 1 X 2…X 2d |v〉, and the symmetric power sum polynomial $\sum_{i=1}^{n} x_{i}^{d}$, for any fixed d>1.
A restriction of determinantal computation is considered in which the underlying affine map λx.L(x) must satisfy a rank lowerability property: L mapping to m×m matrices is said to be r-lowerable, if there exists an $a\in \mbox {$\mathbb {F}$}^{n}$ such that rank (L(a))≤m−r. In this model strongly nonlinear and exponential lower bounds are proved for several polynomial families. For example, for $S^{2d}_{n}$ it is proved that the determinantal complexity using r-lowerable maps is Ω(n d/(2d−r)), for constants d and r with 2≤d+1≤r<2d. For r=2d−1 and d=⌊n 1/5−ε ⌋, a lower bound is given for $S^{2d}_{n}$ of magnitude $n^{\Omega(\epsilon n^{1/5-\epsilon})}$, for any ε∈(0,1/5).
- Lower Bounds for the Determinantal Complexity of Explicit Low Degree Polynomials
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Published: 01 August 2011
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View all- Chillara SMukhopadhyay P(2019)Depth-4 Lower Bounds, Determinantal Complexity: A Unified ApproachComputational Complexity10.1007/s00037-019-00185-428:4(545-572)Online publication date: 1-Dec-2019