TY - GEN

T1 - A polynomial degree bound on equations for non-rigid matrices and small linear circuits

AU - Kumar, Mrinal

AU - Volk, Ben Lee

N1 - Publisher Copyright:
© Mrinal Kumar and Ben Lee Volk.

PY - 2021/2/1

Y1 - 2021/2/1

N2 - We show that there is an equation of degree at most poly(n) for the (Zariski closure of the) set of the non-rigid matrices: that is, we show that for every large enough field F, there is a non-zero n2-variate polynomial P ∈ F[x1,1, . . ., xn,n] of degree at most poly(n) such that every matrix M which can be written as a sum of a matrix of rank at most n/100 and a matrix of sparsity at most n2/100 satisfies P(M) = 0. This confirms a conjecture of Gesmundo, Hauenstein, Ikenmeyer and Landsberg [8] and improves the best upper bound known for this problem down from exp(n2) [11, 8] to poly(n). We also show a similar polynomial degree bound for the (Zariski closure of the) set of all matrices M such that the linear transformation represented by M can be computed by an algebraic circuit with at most n2/200 edges (without any restriction on the depth). As far as we are aware, no such bound was known prior to this work when the depth of the circuits is unbounded. Our methods are elementary and short and rely on a polynomial map of Shpilka and Volkovich [17] to construct low degree “universal” maps for non-rigid matrices and small linear circuits. Combining this construction with a simple dimension counting argument to show that any such polynomial map has a low degree annihilating polynomial completes the proof. As a corollary, we show that any derandomization of the polynomial identity testing problem will imply new circuit lower bounds. A similar (but incomparable) theorem was proved by Kabanets and Impagliazzo [10].

AB - We show that there is an equation of degree at most poly(n) for the (Zariski closure of the) set of the non-rigid matrices: that is, we show that for every large enough field F, there is a non-zero n2-variate polynomial P ∈ F[x1,1, . . ., xn,n] of degree at most poly(n) such that every matrix M which can be written as a sum of a matrix of rank at most n/100 and a matrix of sparsity at most n2/100 satisfies P(M) = 0. This confirms a conjecture of Gesmundo, Hauenstein, Ikenmeyer and Landsberg [8] and improves the best upper bound known for this problem down from exp(n2) [11, 8] to poly(n). We also show a similar polynomial degree bound for the (Zariski closure of the) set of all matrices M such that the linear transformation represented by M can be computed by an algebraic circuit with at most n2/200 edges (without any restriction on the depth). As far as we are aware, no such bound was known prior to this work when the depth of the circuits is unbounded. Our methods are elementary and short and rely on a polynomial map of Shpilka and Volkovich [17] to construct low degree “universal” maps for non-rigid matrices and small linear circuits. Combining this construction with a simple dimension counting argument to show that any such polynomial map has a low degree annihilating polynomial completes the proof. As a corollary, we show that any derandomization of the polynomial identity testing problem will imply new circuit lower bounds. A similar (but incomparable) theorem was proved by Kabanets and Impagliazzo [10].

KW - Circuit lower bounds

KW - Degree bounds

KW - Linear circuits

KW - Rigid matrices

UR - http://www.scopus.com/inward/record.url?scp=85115265649&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ITCS.2021.9

DO - 10.4230/LIPIcs.ITCS.2021.9

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AN - SCOPUS:85115265649

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 12th Innovations in Theoretical Computer Science Conference, ITCS 2021

A2 - Lee, James R.

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 12th Innovations in Theoretical Computer Science Conference, ITCS 2021

Y2 - 6 January 2021 through 8 January 2021

ER -