Deterministic Divisibility Testing via Shifted Partial Derivatives
2015 IEEE 56th Annual Symposium on Foundations of Computer Science
Kayal [Kay12] has recently introduced the method of shifted partial derivatives as a way to give the first exponential lower bound for computing an explicit polynomial as a sum of powers of quadratic polynomials. This method has garnered further attention because of the work of Gupta, Kamath, Kayal and Saptharishi [GKKS14] who used this method to obtain lower bounds that approach the "chasm at depth-4" ([AV08, Koi12, Tav13] ). In this work, we investigate to what extent this method can be used
... o obtain deterministic polynomial identity testing (PIT) algorithms, which are algorithms that decide whether a given algebraic circuit computes the zero polynomial. In particular, we give a poly(s) O(lg s) -time deterministic black-box PIT algorithm for a size-s sum of powers of quadratic polynomials. This is the first sub-exponential deterministic PIT algorithm for this model of computation and the first 1 PIT algorithm based on the method of shifted partial derivatives. We also study the problem of divisibility testing, which when given polynomials f (x) and g(x) (as algebraic circuits) asks to decide whether f (x) divides g(x). Using Strassen's [Str73] technique for the elimination of divisions, we show that one can obtain deterministic divisibility testing algorithms via deterministic PIT algorithms, and this reduction does not dramatically increase the complexity of the underlying algebraic circuits. Using this reduction, we show that deciding divisibility of a quadratic polynomial f into a sparse polynomial g reduces to PIT of sums of a monomial multiplied by a power of quadratic polynomials. We then extend the method of shifted partial derivatives to give a poly(s) O(lg s)time deterministic black-box PIT algorithm for this model of computation, and thus derive a corresponding deterministic divisibility algorithm. This is the first non-trivial deterministic algorithm for this problem. Previous work on multivariate divisibility testing due to Saha, Saptharishi and Saxena [SSS13] gave algorithms for when f is linear and g is sparse, and essentially worked via PIT algorithms for read-once (oblivious) algebraic branching programs (roABPs). We give explicit sums of powers of quadratic polynomials that require exponentially-large roABPs in a strong sense, showing that techniques known for roABPs have limited applicability in our regime. Finally, by combining our results with the algorithm of Forbes, Shpilka and Saptharishi [FSS14] we obtain poly(s) O(lg lg s) -time deterministic black-box PIT algorithms for various models (translations of sparse polynomials, and sums of monomials multiplied by a power of a linear polynomial) where only poly(s) Θ(lg s) -time such algorithms were previously known. We consider two problems from computational algebra: polynomial identity testing (PIT) and divisibility testing. In these problems, one receives algebraic circuits as inputs and the problem is to decide whether the multivariate polynomials that these circuits compute satisfy various properties. That is, the input to these problems is a directed acyclic graph whose internal nodes are labeled with algebraic operations (addition and multiplication) and whose leaves are labeled by variables x i or scalars from a field F. Algebraic circuits naturally compute polynomials in the ring F[x 1 , . . . , x n ] and are one of the most natural and succinct methods to describe such polynomials. However, the succinctness of this description creates challenges in algorithmically understanding the computed polynomial. In particular, while there are often randomized algorithms for deciding properties of the polynomials computed by algebraic circuits, derandomizing such algorithms is an active area of research, in particular because of its connections to derandomizing other well-known problems such as computing perfects matchings in NC ([Lov79, KUW86, MVV87]). Polynomial Identity Testing: The first such problem we consider, polynomial identity testing, asks whether a given algebraic circuit computes the zero polynomial. This problem has a simple randomized algorithm: evaluate the circuit at random point and declare the polynomial nonzero if the evaluation is non-zero. The correctness of this algorithm follows from the Schwartz-Zippel [Sch80, Zip79, DL78] lemma. This randomized algorithm also has the property of being black-box in that it only uses the algebraic circuit as a method to implement an evaluation oracle α → f (α) for the underlying polynomial f (x) computed by this circuit. In contrast, a white-box algorithm is allowed to use the structure of the computation of this circuit. Thus, as the black-box model allows weaker access, deriving results in this model are correspondingly stronger. Creating deterministic PIT algorithms is a significant challenge, as it is known to have implications for the existence of explicit polynomials that require large algebraic circuits for their computation ([HS80, KI04, Agr05]), which is a long standing open question. As such, attention has focused on designing deterministic PIT algorithms for restricted models of algebraic computation. This focus on restricted models has in particular yielded a long line of work for PIT of bounded topfan-in depth-3 and depth-4 circuits [DS07, KS07, KS11, KS09, SS11, KMSV13, SV11, ASSS12, SSS13, SS12, SS13, Gup14]. Recently, this focus has been further justified by depth-reduction results ([VSBR83, AV08, Koi12, Tav13, GKKS13] ) that in particular show that polynomial-time deterministic black-box PIT algorithms for depth-3 algebraic circuits (of exponential degree) imply a deterministic quasipolynomial-time black-box PIT algorithm for general algebraic circuits. These results show that depth-3 circuits essentially capture the full complexity of general algebraic circuits. For more on algebraic circuits, PIT, and depth reduction, see the recent surveys of Shpilka-Yehudayoff [SY10], Saxena [Sax09, Sax14] or Saptharishi [Sap14] . Another direction of study has considered algebraic branching programs (ABPs). As this model can simulate formulas (and thus low depth circuits), this direction of study ([RS05, AMS10, FS12, FS13b, ASS13, FS13a, FSS14, AGKS14, GKST15]) has focused on restricted classes of ABPs such as those that are non-commutative, set-multilinear, or read-once (and oblivious). These three restrictions are essentially the same, and thus we will focus on read-once (oblivious) ABPs (roABPs). This model is partly interesting because it subsumes various other natural models (such as sums of powers of linear forms ([Sax08, FS13b])), and because its complexity is exactly characterized by the rank of Nisan's [Nis91] partial derivative matrix (which as a technique is also used for multilinear formula lower bounds ([Raz06, Raz09, RY09])). Further, developing deterministic polytime black-box PIT has applications to other questions such as derandomizing Noether Normalization ([Mul12, FS13a] ) and can be seen as an algebraic analogue to derandomizing RL (see the discussion in Forbes-1. s-sparse polynomials.