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Dedekind Zeta Zeroes and Faster Complex Dimension Computation∗

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Dedekind Zeta Zeroes and Faster Complex Dimension Computation∗ Dedekind Zeta Zeroes and Faster Complex Dimension Computation∗ J. Maurice Rojas† Yuyu Zhu‡ Texas A&M University Texas A&M University College Station, Texas College Station, Texas [email protected] [email protected] S k ABSTRACT F 2 (Z[x1;:::; xn]) k;n 2Z Thanks to earlier work of Koiran, it is known that the truth of the has a complex root. While the implication FEASC 2 P =) P=NP has Generalized Riemann Hypothesis (GRH) implies that the dimension long been known, the inverse implication FEASC < P =) P , NP of algebraic sets over the complex numbers can be determined remains unknown. Proving the implication FEASC < P =) P,NP within the polynomial-hierarchy. The truth of GRH thus provides a would shed new light on the P vs. NP Problem, and may be easier direct connection between a concrete algebraic geometry problem than attempting to prove the complexity lower bound FEASC < P and the P vs. NP Problem, in a radically different direction from (whose truth is still unknown). the geometric complexity theory approach to VP vs. VNP. We Detecting complex roots is the D =0 case of the following more explore more plausible hypotheses yielding the same speed-up. general problem: One minimalist hypothesis we derive involves improving the error [ DIM ; 2 N × Z ;:::; k term (as a function of the degree, coefficient height, and x) on the C: Given (D F ) ( [x1 xn]) , fraction of primes p ≤ x for which a polynomial has roots mod p. k;n 2Z decide whether the complex zero set of F has A second minimalist hypothesis involves sharpening current zero- dimension at least D. ⋄ free regions for Dedekind zeta functions. Both our hypotheses allow FEAS )DIM failures of GRH but still enable complex dimension computation in In particular, C < P = C < P. Recall the containment of NPNP the polynomial hierarchy. complexity classes P ⊆ NP ⊆ AM ⊆ P ⊆ PSPACE, and that the ? question P = PSPACE remains open [Pap95, AB09]. That P = NP NP CCS CONCEPTS implies the collapse P = NP = coNP = AM = PNP is a basic fact • CCS → Theory of computation, Computational complexity and from complexity theory (see, e.g., [AB09, Thm. 5.4, pp. 97–98]. (We cryptography; Algebraic complexity theory; briefly review these complexity classes in the next section.) DIMC (and thus FEASC) has been known to lie in PSPACE at least since KEYWORDS [GH93], and the underlying algorithms have important precursors polynomial hierarchy, Dedekind zeta, Riemann hypothesis, random in [CG84, Can88, Ren92]. primes, height bounds, rational univariate reduction, nullstellensatz, But in 1996, Koiran [Koi96] proved that the truth of the complex dimension Generalized Riemann Hypothesis (GRH) implies that DIMC 2 AM. In particular, one obtains that the truth of GRH yields the ACM Reference Format: implication DIM P =) P NP. Thus, if one can prove that J. Maurice Rojas and Yuyu Zhu. 2018. Dedekind Zeta Zeroes and Faster C < , Complex Dimension Computation. In Proceedings of ACM International computing the dimension of complex algebraic sets is hard, one can Symposium on Symbolic and Algebraic Computation (ISSAC’18). ACM, New solve the P vs. NP Problem. An interesting application of Koiran’s York, NY, USA, Article ?, 6 pages. https://doi.org/10.475/123_4 result is that it is a key step in the proof that the truth of GRH implies that knottedness (of a curve defined by a knot diagram) can 1 INTRODUCTION be decided in NP [Kup14]. NPNP The subtlety of computational complexity in algebraic geometry persists Here, we prove that DIMC 2 P under either of two new in some of its most basic problems. For instance, let FEASC denote the hypotheses: See Theorem 1 below. Each of our hypotheses is implied problem of deciding whether an input polynomial system by GRH, but can still hold true under certain failures of GRH. Remark. To the best of our knowledge, the only other work on ∗Copyright, 2018 †Partially supported by NSF grants DMS-1460766 and CCF-1409020. improving Koiran’s conditional speed-up has focussed on proving NP ‡Partially supported by NSF grants DMS-1460766 and CCF-1409020. unconditional speed-ups (from PSPACE to PNP or NP) for special families of polynomial systems. See, e.g., [Che07, Roj07]. For instance, Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed thanks to the first paper, the special case of FEASC involving inputs D for profit or commercial advantage and that copies bear this notice and the full citation of the form (f ; x − 1) with (D; f ) 2 N × Z[x1] is NP-complete. ⋄ on the first page. Copyrights for third-party components of this work must be honored. 1 For all other uses, contact the owner/author(s). To state our first and most plausible hypothesis, let f 2 Z[x1] be ISSAC’18, July 2018, New York, USA an irreducible polynomial of degree d with coefficients of absolute © 2018 Copyright held by the owner/author(s). σ ACM ISBN 123-4567-24-567/08/06...$? value at most 2 for some σ 2 N. Let πf (x) denote the number of https://doi.org/10.475/123_4 primes p for which the mod p reduction of f has a root mod p and ISSAC’18, July 2018, New York, USA J. Maurice Rojas and Yuyu Zhu p ≤x. Note that πx1 (x) is thus simply the number of primes p ≤x, We prove Theorem 1 in Section 3. We briefly review some com- i.e., the well-known prime-counting function π (x). In what follows, plexity theoretic notation in Section 2.1, and in Section 2.2 we all O- and Ω- constants are absolute (i.e., they really are constants) review some algebraic tools we need to relate polynomial systems and effectively computable. to number fields. It is important to recall that, like Koiran’s origi- nal approach in [Koi96], our algorithm is completely distinct from Modular Root Hypothesis (MRH). There is a constant C > 1 such that numerical continuation, or the usual computational algebra tech- for any f as above we have niques like Gröbner bases, resultants, or non-Archimedean Newton 1 1 Iteration. In particular, we use random sampling to study the den- π (x) ≥ x − . f d log x (log x )1=C exp sity of primes p for which the mod p reduction of a polynomial (log(d2σ +d3))C * 2 + system has roots over the finite field F . for x = Ω exp 4(log(d2.σ + d3))2+C . / p , - That π (x) is asymptotic to x for some positive integer s ≤d 2 TECHNICAL BACKGROUND f sf log x f goes back to classical work of Frobenius [Fro96] (see also [LS96] Our approach begins by naturally associating a number field K for an excellent historical discussion). More to the point, as we’ll to a polynomial system F = (f1;:::; fk ) 2 Z[x1;:::; xn]. Then, the see in our proofs, the behavior of πf is intimately related to the distribution of prime ideals of OK forces the existence of complex distribution of prime ideals in the ring OK of algebraic integers in roots for F to imply (unconditionally) the existence of roots over the number field K :=Q[x1]=hf i, and the error term is where all the Fp for a positive density of mod p reductions of F. Conversely, if F difficulty enters: MRH is not currently known to be true. However, has no complex roots, then there are (unconditionally) only finitely MRH can still hold even if GRH fails (see Theorem 1 below). In many primes p such that the mod p reduction of F has a root over particular, while the truth of GRH implies that the 1=exponential Fp . These observations, along with a clever random-sampling trick d (∆x ) term in our lower bound above can be decreased to O logp in that formed the first algorithm for computing complex dimension x in the polynomial-hierarchy (assuming GRH), go back to Koiran absolute value, we will see later that our looser bound still suffices [Koi96]. Our key contribution is thus isolating the minimal number- for our algorithmic purposes. (Note that p1 =o 1 for x exp((log x )1=C ) theoretic hypotheses (“strictly” more plausible than GRH) sufficient any C > 1.) to make a positive density of primes observable via efficient random Our second hypothesis is a statement intermediate between MRH sampling. and GRH in plausibility. Recall that the Dedekind zeta function, ζK (s), is the analytic continuation (to C n f1g) of the 2.1 Some Complexity Theory P 1 function s , where the summation is taken over all a (N a) Our underlying computational model will be the classical Turing integral ideals a of OK and Na is the norm of a [IK04]. (So ζQ(s) machine, which, informally, can be assumed to be anyone’s laptop is the classical Riemann zeta function ζ (s), defined from the sum computer, augmented with infinite memory and a flawless operat- 1 p P 1 ing system. Our notion of input size is the following: ns .) We call a root ρ = β + γ −1 (with β;γ 2 R) of ζK a n=1 non-trivial zero if and only if 0 < β < 1. GRH is then following Definition 1. The bit-size (or sparse size) of a polynomial sys- statement: tem F B (f1; ··· ; fk ) 2 Z[x1;:::; xn], is defined to be the total p number of bits in the binary expansions of all the coefficients and (GRH) All the non-trivial zeroes ρ = β + γ −1 of ζK lie on the vertical line defined by β = 1=2.
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