scholarly journals Factorization of Polynomials Given by Arithmetic Branching Programs

2021 ◽  
Vol 30 (2) ◽  
Author(s):  
Amit Sinhababu ◽  
Thomas Thierauf
Keyword(s):  
Upper Bound ◽  
Arithmetic Circuits ◽  
Branching Programs ◽  
Multivariate Polynomial ◽  
Factorization Of Polynomials

AbstractGiven a multivariate polynomial computed by an arithmetic branching program (ABP) of size s, we show that all its factors can be computed by arithmetic branching programs of size poly(s). Kaltofen gave a similar result for polynomials computed by arithmetic circuits. The previously known best upper bound for ABP-factors was poly $$ (s^{ {\rm \log} s}) $$ ( s log s ) .

scholarly journals On Zeros of a Polynomial in a Finite Grid

2018 ◽  
Vol 27 (3) ◽  
pp. 310-333 ◽  
Author(s):  
ANURAG BISHNOI ◽  
PETE L. CLARK ◽  
ADITYA POTUKUCHI ◽  
JOHN R. SCHMITT
Keyword(s):  
Upper Bound ◽  
Integral Domain ◽  
Hamming Distance ◽  
Finite Geometry ◽  
Blocking Sets ◽  
Multivariate Polynomial ◽  
Number Of Zeros ◽  
Minimum Hamming Distance ◽  
Affine Variety Codes ◽  
The Relationship

A 1993 result of Alon and Füredi gives a sharp upper bound on the number of zeros of a multivariate polynomial over an integral domain in a finite grid, in terms of the degree of the polynomial. This result was recently generalized to polynomials over an arbitrary commutative ring, assuming a certain ‘Condition (D)’ on the grid which holds vacuously when the ring is a domain. In the first half of this paper we give a further generalized Alon–Füredi theorem which provides a sharp upper bound when the degrees of the polynomial in each variable are also taken into account. This yields in particular a new proof of Alon–Füredi. We then discuss the relationship between Alon–Füredi and results of DeMillo–Lipton, Schwartz and Zippel. A direct coding theoretic interpretation of Alon–Füredi theorem and its generalization in terms of Reed–Muller-type affine variety codes is shown, which gives us the minimum Hamming distance of these codes. Then we apply the Alon–Füredi theorem to quickly recover – and sometimes strengthen – old and new results in finite geometry, including the Jamison–Brouwer–Schrijver bound on affine blocking sets. We end with a discussion of multiplicity enhancements.

On Proving Parameterized Size Lower Bounds for Multilinear Algebraic Models

2020 ◽  
Vol 177 (1) ◽  
pp. 69-93
Author(s):  
Purnata Ghosal ◽  
B.V. Raghavendra Rao
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Partial Derivative ◽  
Computable Function ◽  
Arithmetic Circuits ◽  
Multilinear Polynomial ◽  
Algebraic Models ◽  
Branching Programs ◽  
Algebraic Branching Programs ◽  
Derivative Matrix

We consider the problem of obtaining parameterized lower bounds for the size of arithmetic circuits computing polynomials with the degree of the polynomial as the parameter. We consider the following special classes of multilinear algebraic branching programs: 1) Read Once Oblivious Branching Programs (ROABPs), 2) Strict interval branching programs, 3) Sum of read once formulas with restricted ordering. We obtain parameterized lower bounds (i.e., nΩ(t(k)) lower bound for some function t of k) on the size of the above models computing a multilinear polynomial that can be computed by a depth four circuit of size g(k)nO(1) for some computable function g. Further, we obtain a parameterized separation between ROABPs and read-2 ABPs. This is obtained by constructing a degree k polynomial that can be computed by a read-2 ABP of small size such that the rank of the partial derivative matrix under any partition of the variables is large.

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