A Lower Bound for the Formula Size of Rational Functions

Kalorkoti, K.

doi:10.1137/0214050

Cited by 44 publications

(22 citation statements)

References 4 publications

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“…Two reasons behind this generalization are: One, to accommodate the power of exponentiation -if we take the e-th power of a read-k formula using a product gate, the 'read' of the resulting formula goes up to ek -we would like to avoid this superfluous blow up in read. Two, a read-k formula has size O(kn), which severely hinders its power of computation -for instance, determinant and permanent cannot even be expressed in this model when k is a constant [Kal85]. This calls for the following definition.…”

Section: A Tale Of Two Pits (And Three Lower Bounds)mentioning

confidence: 99%

Jacobian hits circuits

Agrawal

Saha

Saptharishi

et al. 2012

Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing

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We present a single common tool to strictly subsume all known cases of polynomial time blackbox polynomial identity testing (PIT), that have been hitherto solved using diverse tools and techniques, over fields of zero or large characteristic. In particular, we show that polynomial time hitting-set generators for identity testing of the two seemingly different and well studied models -depth-3 circuits with bounded top fanin, and constant-depth constant-read multilinear formulas -can be constructed using one common algebraic-geometry theme: Jacobian captures algebraic independence. By exploiting the Jacobian, we design the first efficient hitting-set generators for broad generalizations of the above-mentioned models, namely:

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Section: A Tale Of Two Pits (And Three Lower Bounds)mentioning

confidence: 99%

Jacobian hits circuits

Agrawal

Saha

Saptharishi

et al. 2012

Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing

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“…The absence of substantial progress on this general question has led to focus on the question of proving better lower bounds for restricted and more structured subclasses of arithmetic circuits. Arithmetic formulas [Kal85], non-commutative arithmetic circuits [Nis91], algebraic branching programs [Kum17], and low depth arithmetic circuits [NW97, GK98, GR00, Raz10, GKKS14, FLMS14, KLSS14, KS14,KS17] are some such subclasses which have been studied from this perspective. For an overview of the definition of these models and the state of art for lower bounds for them, we refer the reader to the surveys of Shpilka and Yehudayoff [SY10] and Saptharishi [Sap16].…”

Section: Introductionmentioning

confidence: 99%

Unbalancing Sets and An Almost Quadratic Lower Bound for Syntactically Multilinear Arithmetic Circuits

2020

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We prove a lower bound of Ω(n 2 / log 2 n) on the size of any syntactically multilinear arithmetic circuit computing some explicit multilinear polynomial f (x 1 , . . . , x n ). Our approach expands and improves upon a result of Raz, Shpilka and Yehudayoff ([RSY08]), who proved a lower bound of Ω(n 4/3 / log 2 n) for the same polynomial. Our improvement follows from an asymptotically optimal lower bound for a generalized version of Galvin's problem in extremal set theory.

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“…We believe that finding a Jacobian-type polynomial that captures algebraic independence in any characteristic p > 0 is a natural question in algebra and geometry. Furthermore, Jacobian has recently found several applications in complexity theorycircuit lower bound proofs [Kal85,ASSS12], pseudo-random objects construction [DGW09,Dvi09], identity testing [BMS11,ASSS12], cryptography [DGRV11], program invariants [L'v84, Kay09], and control theory [For91,DF92]. Thus, a suitably effective Jacobian-type criterion is desirable to make these applications work for any field.…”

Section: Introductionmentioning

confidence: 99%

Algebraic independence in positive characteristic: A $p$-adic calculus

Mittmann¹,

Saxena

Scheiblechner

2014

Trans. Amer. Math. Soc.

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Abstract. A set of multivariate polynomials, over a field of zero or large characteristic, can be tested for algebraic independence by the well-known Jacobian criterion. For fields of other characteristic p > 0, there is no analogous characterization known. In this paper we give the first such criterion. Essentially, it boils down to a non-degeneracy condition on a lift of the Jacobian polynomial over (an unramified extension of) the ring of p-adic integers.Our proof builds on the de Rham-Witt complex, which was invented by Illusie (1979) for crystalline cohomology computations, and we deduce a natural generalization of the Jacobian. This new avatar we call the Witt-Jacobian. In essence, we show how to faithfully differentiate polynomials over Fp (i.e. somehow avoid ∂x p /∂x = 0) and thus capture algebraic independence.We apply the new criterion to put the problem of testing algebraic independence in the complexity class NP #P (previously best was PSPACE). Also, we give a modest application to the problem of identity testing in algebraic complexity theory.

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A Lower Bound for the Formula Size of Rational Functions

Cited by 44 publications

References 4 publications

Jacobian hits circuits

Jacobian hits circuits

Unbalancing Sets and An Almost Quadratic Lower Bound for Syntactically Multilinear Arithmetic Circuits

Algebraic independence in positive characteristic: A $p$-adic calculus

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