Abstract
The motivation for this work (Pandey et al. 2016) comes from two problems: testing algebraic independence of arithmetic circuits over a field of small characteristic and generalizing the structural property of algebraic dependence used by Kumar, Saraf, CCC’16 to arbitrary fields. It is known that in the case of zero, or large characteristic, using a classical criterion based on the Jacobian, we get a randomized poly-time algorithm to test algebraic independence. Over small characteristic, the Jacobian criterion fails and there is no subexponential time algorithm known. This problem could well be conjectured to be in RP, but the current best algorithm puts it in NP $${^{\#{\rm P}}}$$ (Mittmann, Saxena, Scheiblechner, Trans.AMS’14). Currently, even the case of two bivariate circuits over $${\mathbb{F}_2}$$ is open. We come up with a natural generalization of Jacobian criterion that works over all characteristics. The new criterion is efficient if the underlying inseparable degree is promised to be a constant. This is a modest step toward the open question of fast independence testing, over finite fields, posed in (Dvir, Gabizon, Wigderson, FOCS’07). In a set of linearly dependent polynomials, any polynomial can be written as a linear combination of the polynomials forming a basis. The analogous property for algebraic dependence is false, but a property approximately in that spirit is named as “functional dependence” in Kumar, Saraf, CCC’16 and proved for zero or large characteristics. We show that functional dependence holds for arbitrary fields, thereby answering the open questions in Kumar, Saraf, CCC’16. Following them, we use the functional dependence lemma to prove the first exponential lower bound for locally low algebraic rank circuits for arbitrary fields (a model that strongly generalizes homogeneous depth-4 circuits). We also recover their quasipoly-time hitting-set for such models, for fields of characteristic smaller than the ones known before. Our results show that approximate functional dependence is indeed a more fundamental concept than the Jacobian as it is field independent. We achieve the former by first picking a “good” transcendence basis, then translating the circuits by new variables, and finally approximating them by truncating higher degree monomials. We give a tight analysis of the “degree” of approximation needed in the criterion. To get the locally low-algebraic-rank circuit applications, we follow the known shifted partial derivative-based methods.
Highlights
Algebraic dependence is a fundamental concept in algebra that captures algebraic/polynomial relationship of objects like numbers, polynomials, rational functions or power series, over some field
The key idea is a classical result, known as the Jacobian criterion [20, 4]. It says that if the characteristic of the field is zero, or large enough, the transcendence degree equals the linear rank of the Jacobian matrix of the polynomials. This leads to a simple randomized poly-time algorithm for checking algebraic independence, as we can get the circuits of the partial derivatives efficiently [3] and use random evaluations to compute the rank of the Jacobian matrix
In the world of arithmetic complexity, we have strong structural results like depth reductions [16, 2]. These results show that strong enough lower bound, or PIT, results for homogeneous depth-4 circuits would give us exponential lower bounds and quasipoly-time derandomized PIT for general circuits
Summary
Algebraic dependence is a fundamental concept in algebra that captures algebraic/polynomial relationship of objects like numbers, polynomials, rational functions or power series, over some field. It says that if the characteristic of the field is zero, or large enough (compared to the product of degrees of the given polynomials), the transcendence degree equals the linear rank of the Jacobian matrix of the polynomials This leads to a simple randomized poly-time algorithm for checking algebraic independence, as we can get the circuits of the partial derivatives efficiently [3] and use random evaluations to compute the rank of the Jacobian matrix. Computational implications of an efficient Jacobian like criterion would include a possible generalization (to small characteristic) of PIT or lower bound results [1], and algebraic extractors or entropy concepts [7]. Our goal in this paper is to overcome such restrictions
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