An official website of the United States government
On the Rational Degree of Boolean Functions and Applications Vishnu Iyer, Siddhartha Jain, Matt Kovacs-Deak, Vinayak M. Kumar, Luke Schaeffer, Daochen Wang, Michael Whitmeyer We study a natural complexity measure of Boolean functions known as the (exact) rational degree. For total functions f, it is conjectured that rdeg(f) is polynomially related to deg(f), where deg(f) is the Fourier degree. Towards this conjecture, we show that symmetric functions have rational degree at least deg(f)/2 and monotone functions have rational degree at least sqrt(deg(f)). We observe that both of these lower bounds are tight. In addition, we show that all read-once depth-d Boolean formulae have rational degree at least Ω(deg(f)1/d). Furthermore, we show that almost every Boolean function on n variables has rational degree at least n/2−O(sqrt(n)). In contrast to total functions, we exhibit partial functions that witness unbounded separations between rational and approximate degree, in both directions. As a consequence, we show that for quantum computers, post-selection and bounded-error are incomparable resources in the black-box model.
more »
« less
Fast Multivariate Multipoint Evaluation Over All Finite Fields
Multivariate multipoint evaluation is the problem of evaluating a multivariate polynomial, given as a coefficient vector, simultaneously at multiple evaluation points. In this work, we show that there exists a deterministic algorithm for multivariate multipoint evaluation over any finite field F that outputs the evaluations of an m-variate polynomial of degree less than d in each variable at N points in time (dm + N)1+o(1) · poly(m, d, log |F|) for all m ∈ N and all sufficiently large d ∈ N. A previous work of Kedlaya and Umans (FOCS 2008, SICOMP 2011) achieved the same time complexity when the number of variables m is at most d^{o(1)} and had left the problem of removing this condition as an open problem. A recent work of Bhargava, Ghosh, Kumar and Mohapatra (STOC 2022) answered this question when the underlying field is not too large and has characteristic less than d^{o(1)}. In this work, we remove this constraint on the number of variables over all finite fields, thereby answering the question of Kedlaya and Umans over all finite fields. Our algorithm relies on a non-trivial combination of ideas from three seemingly different previously knownalgorithms for multivariate multipoint evaluation, namely the algorithms of Kedlaya and Umans, that of Björklund, Kaski and Williams (IPEC 2017, Algorithmica 2019), and that of Bhargava, Ghosh, Kumar and Mohapatra, together with a result of Bombieri and Vinogradov from analytic number theory about the distribution of primes in an arithmetic progression. We also present a second algorithm for multivariate multipoint evaluation that is completely elementary and in particular, avoids the use of the Bombieri–Vinogradov Theorem. However, it requires a mild assumption that the field size is bounded by an exponential-tower in d of bounded height.
more »
« less
- Award ID(s):
- 1909683
- PAR ID:
- 10466778
- Publisher / Repository:
- IEEE
- Date Published:
- ISBN:
- 978-1-6654-5519-0
- Page Range / eLocation ID:
- 221 to 232
- Format(s):
- Medium: X
- Location:
- Denver, CO, USA
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
We prove that the most natural low-degree test for polynomials over finite fields is “robust” in the high-error regime for linear-sized fields. Specifically we consider the “local” agreement of a function $$f:\mathbb{F}_{q}^{m}\rightarrow \mathbb{F}_{q}$$ from the space of degree-d polynomials, i.e., the expected agreement of the function from univariate degree-d polynomials over a randomly chosen line in $$\mathbb{F}_{q}^{m}$$, and prove that if this local agreement is $$\varepsilon\geq\Omega((d/q)^{\tau}))$$ for some fixed $$\tau > 0$$, then there is a global degree-d polynomial $$Q:\mathbb{F}_{q}^{m}\rightarrow \mathbb{F}_{q}$$ with agreement nearly $$\varepsilon$$ with $$f$$. This settles a long-standing open question in the area of low-degree testing, yielding an $O(d)$ -query robust test in the “high-error” regime (i.e., when $$\varepsilon < 1/2)$$. The previous results in this space either required $$\varepsilon > 1/2$$ (Polishchuk & Spielman, STOC 1994), or $$q=\Omega(d^{4})$$ (Arora & Sudan, Combinatorica 2003), orneeded to measure local distance on 2-dimensional “planes” rather than one-dimensional lines leading to $$\Omega(d^{2})$$ -query complexity (Raz & Safra, STOC 1997). Our analysis follows the spirit of most previous analyses in first analyzing the low-variable case $(m=O(1))$ and then “boot-strapping” to general multivariate settings. Our main technical novelty is a new analysis in the bivariate setting that exploits a previously known connection between multivariate factorization and finding (or testing) low-degree polynomials, in a non “black-box” manner. This connection was used roughly in a black-box manner in the work of Arora & Sudan — and we show that opening up this black box and making some delicate choices in the analysis leads to our essentially optimal analysis. A second contribution is a bootstrapping analysis which manages to lift analyses for $m=2$ directly to analyses for general $$m$$, where previous works needed to work with $m=3$ or $m=4$ — arguably this bootstrapping is significantly simpler than those in prior works.more » « less
-
We prove that the most natural low-degree test for polynomials over finite fields is “robust” in the high-error regime for linear-sized fields. Specifically we consider the “local” agreement of a function $$f:\mathbb{F}_{q}^{m}\rightarrow \mathbb{F}_{q}$$ from the space of degree-d polynomials, i.e., the expected agreement of the function from univariate degree-d polynomials over a randomly chosen line in $$\mathbb{F}_{q}^{m}$$, and prove that if this local agreement is $$\varepsilon\geq\Omega((d/q)^{\tau}))$$ for some fixed $$\tau > 0$$, then there is a global degree-d polynomial $$Q:\mathbb{F}_{q}^{m}\rightarrow \mathbb{F}_{q}$$ with agreement nearly $$\varepsilon$$ with $$f$$. This settles a long-standing open question in the area of low-degree testing, yielding an $O(d)$ -query robust test in the “high-error” regime (i.e., when $$\varepsilon < 1/2)$$. The previous results in this space either required $$\varepsilon > 1/2$$ (Polishchuk & Spielman, STOC 1994), or $$q=\Omega(d^{4})$$ (Arora & Sudan, Combinatorica 2003), orneeded to measure local distance on 2-dimensional “planes” rather than one-dimensional lines leading to $$\Omega(d^{2})$$ -query complexity (Raz & Safra, STOC 1997). Our analysis follows the spirit of most previous analyses in first analyzing the low-variable case $(m=O(1))$ and then “boot-strapping” to general multivariate settings. Our main technical novelty is a new analysis in the bivariate setting that exploits a previously known connection between multivariate factorization and finding (or testing) low-degree polynomials, in a non “black-box” manner. This connection was used roughly in a black-box manner in the work of Arora & Sudan — and we show that opening up this black box and making some delicate choices in the analysis leads to our essentially optimal analysis. A second contribution is a bootstrapping analysis which manages to lift analyses for $m=2$ directly to analyses for general $$m$$, where previous works needed to work with $m=3$ or $m=4$ — arguably this bootstrapping is significantly simpler than those in prior works.more » « less
-
Peter Burgisser (Ed.)Suppose A = {a 1 , . . . , a n+2 } ⊂ Z n has cardinality n + 2, with all the coordinates of the a j having absolute value at most d, and the a j do not all lie in the same affine hyperplane. Suppose F = ( f 1 , . . . , f n ) is an n × n polynomial system with generic integer coefficients at most H in absolute value, and A the union of the sets of exponent vectors of the f i . We give the first algorithm that, for any fixed n, counts exactly the number of real roots of F in time polynomial in log(dH ). We also discuss a number- theoretic hypothesis that would imply a further speed-up to time polynomial in n as well.more » « less
-
Consider a system of m polynomial equations {pi(x)=bi}i≤m of degree D≥2 in n-dimensional variable x∈ℝn such that each coefficient of every pi and bis are chosen at random and independently from some continuous distribution. We study the basic question of determining the smallest m -- the algorithmic threshold -- for which efficient algorithms can find refutations (i.e. certificates of unsatisfiability) for such systems. This setting generalizes problems such as refuting random SAT instances, low-rank matrix sensing and certifying pseudo-randomness of Goldreich's candidate generators and generalizations. We show that for every d∈ℕ, the (n+m)O(d)-time canonical sum-of-squares (SoS) relaxation refutes such a system with high probability whenever m≥O(n)⋅(nd)D−1. We prove a lower bound in the restricted low-degree polynomial model of computation which suggests that this trade-off between SoS degree and the number of equations is nearly tight for all d. We also confirm the predictions of this lower bound in a limited setting by showing a lower bound on the canonical degree-4 sum-of-squares relaxation for refuting random quadratic polynomials. Together, our results provide evidence for an algorithmic threshold for the problem at m≳O˜(n)⋅n(1−δ)(D−1) for 2nδ-time algorithms for all δ.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/FOCS54457.2022.00028
