Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: 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}$}$.#R##N##R##N#Asymptotically tight lor 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 por sum polynomial $\sum_{i=1}^{n} x_{i}^{d}$, for any fixed d>1.#R##N##R##N#A restriction of determinantal computation is considered in which the underlying affine map λx.L(x) must satisfy a rank lorability property: L mapping to m×m matrices is said to be r-lorable, if there exists an $a\in \mbox {$\mathbb {F}$}^{n}$ such that rank (L(a))≤m−r. In this model strongly nonlinear and exponential lor bounds are proved for several polynomial families. For example, for $S^{2d}_{n}$ it is proved that the determinantal complexity using r-lorable 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−e ⌋, a lor bound is given for $S^{2d}_{n}$ of magnitude $n^{\Omega(\epsilon n^{1/5-\epsilon})}$, for any e∈(0,1/5).
['Maurice J. Jansen']
I am writing a research paper on a idea Lower Bounds for the Determinantal Complexity of Explicit Low Degree Polynomials
[{"from": "human", "value": "I am writing a research paper on a idea Lower Bounds for the Determinantal Complexity of Explicit Low Degree Polynomials"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The determinantal complexity of a polynomial f(x 1,x 2,\u2026,x n ) is the minimum m such that f=det\u2009m (L(x 1,x 2,\u2026,x n )), where L(x 1,x 2,\u2026,x n ) is a matrix whose entries are affine forms in the x i s over some field $\\mbox {$\\mathbb {F}$}$.#R##N##R##N#Asymptotically tight lor 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 \u2329u|X 1 X 2\u2026X 2d |v\u232a, and the symmetric por sum polynomial $\\sum_{i=1}^{n} x_{i}^{d}$, for any fixed d>1.#R##N##R##N#A restriction of determinantal computation is considered in which the underlying affine map \u03bbx.L(x) must satisfy a rank lorability property: L mapping to m\u00d7m matrices is said to be r-lorable, if there exists an $a\\in \\mbox {$\\mathbb {F}$}^{n}$ such that rank\u2009(L(a))\u2264m\u2212r. In this model strongly nonlinear and exponential lor bounds are proved for several polynomial families. For example, for $S^{2d}_{n}$ it is proved that the determinantal complexity using r-lorable maps is \u03a9(n d/(2d\u2212r)), for constants d and r with 2\u2264d+1\u2264r<2d. For r=2d\u22121 and d=\u230an 1/5\u2212e \u230b, a lor bound is given for $S^{2d}_{n}$ of magnitude $n^{\\Omega(\\epsilon n^{1/5-\\epsilon})}$, for any e\u2208(0,1/5)."}]