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### Revision(s):

Revision #2 to TR14-003 | 19th February 2014 06:05

#### Testing Equivalence of Polynomials under Shifts

Revision #2
Authors: Zeev Dvir, Rafael Mendes de Oliveira, Amir Shpilka
Accepted on: 19th February 2014 06:05
Keywords:

Abstract:

Two polynomials $f, g \in F[x_1, \ldots, x_n]$ are called shift-equivalent if there exists a vector $(a_1, \ldots, a_n) \in {F}^n$ such that the polynomial identity $f(x_1+a_1, \ldots, x_n+a_n) \equiv g(x_1,\ldots,x_n)$ holds. Our main result is a new randomized algorithm that tests whether two given polynomials are shift equivalent. Our algorithm runs in time polynomial in the circuit size of the polynomials, to which it is given black box access. This complements a previous work of Grigoriev (Theoretical Computer Science, 1997) who gave a deterministic algorithm running in time $n^{O(d)}$ for degree $d$ polynomials.

Our algorithm uses randomness only to solve instances of the Polynomial Identity Testing (PIT) problem. Hence, if one could de-randomize PIT (a long-standing open problem in complexity) a de-randomization of our algorithm would follow. This establishes an equivalence between de-randomizing shift-equivalence testing and de-randomizing PIT (both in the black-box and the white-box setting). For certain restricted models, such as Read Once Branching Programs, we already obtain a deterministic algorithm using existing PIT results.

Changes to previous version:

and added alternative approach in the appendix.

Revision #1 to TR14-003 | 16th January 2014 17:42

#### Testing Equivalence of Polynomials under Shifts

Revision #1
Authors: Zeev Dvir, Rafael Mendes de Oliveira, Amir Shpilka
Accepted on: 16th January 2014 17:42
Keywords:

Abstract:

Two polynomials $f, g \in F[x_1, \ldots, x_n]$ are called shift-equivalent if there exists a vector $(a_1, \ldots, a_n) \in {F}^n$ such that the polynomial identity $f(x_1+a_1, \ldots, x_n+a_n) \equiv g(x_1,\ldots,x_n)$ holds. Our main result is a new randomized algorithm that tests whether two given polynomials are shift equivalent. Our algorithm runs in time polynomial in the circuit size of the polynomials, to which it is given black box access. This complements a previous work of Grigoriev (Theoretical Computer Science, 1997) who gave a deterministic algorithm running in time $n^{O(d)}$ for degree $d$ polynomials.

Our algorithm uses randomness only to solve instances of the Polynomial Identity Testing (PIT) problem. Hence, if one could de-randomize PIT (a long-standing open problem in complexity) a de-randomization of our algorithm would follow. This establishes an equivalence between de-randomizing shift-equivalence testing and de-randomizing PIT (both in the black-box and the white-box setting). For certain restricted models, such as Read Once Branching Programs, we already obtain a deterministic algorithm using existing PIT results.

Changes to previous version:

Fixed a typo

### Paper:

TR14-003 | 10th January 2014 08:58

#### Testing Equivalence of Polynomials under Shifts

TR14-003
Authors: Zeev Dvir, Rafael Mendes de Oliveira, Amir Shpilka
Publication: 10th January 2014 09:00
Keywords:

Abstract:

Two polynomials $f, g \in F[x_1, \ldots, x_n]$ are called shift-equivalent if there exists a vector $(a_1, \ldots, a_n) \in {F}^n$ such that the polynomial identity $f(x_1+a_1, \ldots, x_n+a_n) \equiv g(x_1,\ldots,x_n)$ holds. Our main result is a new randomized algorithm that tests whether two given polynomials are shift equivalent. Our algorithm runs in time polynomial in the circuit size of the polynomials, to which it is given black box access. This complements a previous work of Grigoriev (Theoretical Computer Science, 1997) who gave a deterministic algorithm running in time $n^{O(d)}$ for degree $d$ polynomials.

Our algorithm uses randomness only to solve instances of the Polynomial Identity Testing (PIT) problem. Hence, if one could de-randomize PIT (a long-standing open problem in complexity) a de-randomization of our algorithm would follow. This establishes an equivalence between de-randomizing shift-equivalence testing and de-randomizing PIT (both in the black-box and the white-box setting). For certain restricted models, such as Read Once Branching Programs, we already obtain a deterministic algorithm using existing PIT results.

### Comment(s):

Comment #1 to TR14-003 | 15th May 2014 09:56

#### Theorem number

Authors: Gorav Jindal
Accepted on: 15th May 2014 09:56
Keywords:

Comment:

On Page 3, it should be "see Theorem 4.1 of [SY10] for a proof" instead of see "Theorem 4.3 of [SY10] for a proof".

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