Abstract Let f : ℙ 1 → ℙ 1 {f:\mathbb{P}^{1}\to\mathbb{P}^{1}} be a map of degree > 1 {>1} defined over a function field k = K ( X ) {k=K(X)} , where K is a number field and X is a projective curve over K . For each point a ∈ ℙ 1 ( k ) {a\in\mathbb{P}^{1}(k)} satisfying a dynamical stability condition, we prove that the Call–Silverman canonical height for specialization f t {f_{t}} at point a t {a_{t}} , for t ∈ X ( ℚ ¯ ) {t\in X(\overline{\mathbb{Q}})} outside a finite set, induces a Weil height on the curve X ; i.e., we prove the existence of a ℚ {\mathbb{Q}} divisor D = D f , a {D=D_{f,a}} on X so that the function t ↦ h ^ f t ( a t )  h D ( t ) {t\mapsto\hat{h}_{f_{t}}(a_{t})h_{D}(t)} is bounded on X ( ℚ ¯ ) {X(\overline{\mathbb{Q}})} for any choice of Weil height associated to D . We also prove a local version, that the local canonical heights t ↦ λ ^ f t , v ( a t ) {t\mapsto\hat{\lambda}_{f_{t},v}(a_{t})} differ from a Weil function for D by a continuous function on X ( ℂ v ) {X(\mathbb{C}_{v})} , at each place v of the number field K . These results were known for polynomial maps f and all points a ∈ ℙ 1 ( k ) {a\in\mathbb{P}^{1}(k)} without the stability hypothesis,[21, 14],and for maps f that are quotients of endomorphisms of elliptic curves E over k and all points a ∈ ℙ 1 ( k ) {a\in\mathbb{P}^{1}(k)} . [32, 29].Finally, we characterize our stability condition in terms of the geometry of the induced map f ~ : X × ℙ 1 ⇢ X × ℙ 1 {\tilde{f}:X\times\mathbb{P}^{1}\dashrightarrow X\times\mathbb{P}^{1}} over K ; and we prove the existence of relative Néron models for the pair ( f , a ) {(f,a)} , when a is a Fatou point at a place γ of k , where the local canonical height λ ^ f , γ ( a ) {\hat{\lambda}_{f,\gamma}(a)} can be computed as an intersection number.
more »
« less
If You Must Choose Among Your Children, Pick the Right One
Given a simplicial complex K and an injective function f from the vertices of K to R, we consider algorithms that extend f to a discrete Morse function on K. We show that an algorithm of King, Knudson and Mramor can be described on the directed Hasse diagram of K. Our description has a faster runtime for high dimensional data with no increase in space.
more »
« less
 NSFPAR ID:
 10309897
 Date Published:
 Journal Name:
 Proceedings of the 32nd Canadian Conference on Computational Geometry (CCCG 2020)
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this


The epsilonapproximate degree, deg_epsilon(f), of a Boolean function f is the least degree of a realvalued polynomial that approximates f pointwise to within epsilon. A sound and complete certificate for approximate degree being at least k is a pair of probability distributions, also known as a dual polynomial, that are perfectly kwise indistinguishable, but are distinguishable by f with advantage 1  epsilon. Our contributions are:  We give a simple, explicit new construction of a dual polynomial for the AND function on n bits, certifying that its epsilonapproximate degree is Omega (sqrt{n log 1/epsilon}). This construction is the first to extend to the notion of weighted degree, and yields the first explicit certificate that the 1/3approximate degree of any (possibly unbalanced) readonce DNF is Omega(sqrt{n}). It draws a novel connection between the approximate degree of AND and anticoncentration of the Binomial distribution.  We show that any pair of symmetric distributions on nbit strings that are perfectly kwise indistinguishable are also statistically Kwise indistinguishable with at most K^{3/2} * exp (Omega (k^2/K)) error for all k < K <= n/64. This bound is essentially tight, and implies that any symmetric function f is a reconstruction function with constant advantage for a ramp secret sharing scheme that is secure against sizeK coalitions with statistical error K^{3/2} * exp (Omega (deg_{1/3}(f)^2/K)) for all values of K up to n/64 simultaneously. Previous secret sharing schemes required that K be determined in advance, and only worked for f=AND. Our analysis draws another new connection between approximate degree and concentration phenomena. As a corollary of this result, we show that for any d <= n/64, any degree d polynomial approximating a symmetric function f to error 1/3 must have coefficients of l_1norm at least K^{3/2} * exp ({Omega (deg_{1/3}(f)^2/d)}). We also show this bound is essentially tight for any d > deg_{1/3}(f). These upper and lower bounds were also previously only known in the case f=AND.more » « less

Chakrabarti, Amit ; Swamy, Chaitanya (Ed.)A Boolean maximum constraint satisfaction problem, MaxCSP(f), is specified by a predicate f:{1,1}^k → {0,1}. An nvariable instance of MaxCSP(f) consists of a list of constraints, each of which applies f to k distinct literals drawn from the n variables. For k = 2, Chou, Golovnev, and Velusamy [Chou et al., 2020] obtained explicit ratios characterizing the √ nspace streaming approximability of every predicate. For k ≥ 3, Chou, Golovnev, Sudan, and Velusamy [Chou et al., 2022] proved a general dichotomy theorem for √ nspace sketching algorithms: For every f, there exists α(f) ∈ (0,1] such that for every ε > 0, MaxCSP(f) is (α(f)ε)approximable by an O(log n)space linear sketching algorithm, but (α(f)+ε)approximation sketching algorithms require Ω(√n) space. In this work, we give closedform expressions for the sketching approximation ratios of multiple families of symmetric Boolean functions. Letting α'_k = 2^{(k1)} (1k^{2})^{(k1)/2}, we show that for odd k ≥ 3, α(kAND) = α'_k, and for even k ≥ 2, α(kAND) = 2α'_{k+1}. Thus, for every k, kAND can be (2o(1))2^{k}approximated by O(log n)space sketching algorithms; we contrast this with a lower bound of Chou, Golovnev, Sudan, Velingker, and Velusamy [Chou et al., 2022] implying that streaming (2+ε)2^{k}approximations require Ω(n) space! We also resolve the ratio for the "atleast(k1)1’s" function for all even k; the "exactly(k+1)/21’s" function for odd k ∈ {3,…,51}; and fifteen other functions. We stress here that for general f, the dichotomy theorem in [Chou et al., 2022] only implies that α(f) can be computed to arbitrary precision in PSPACE, and thus closedform expressions need not have existed a priori. Our analyses involve identifying and exploiting structural "saddlepoint" properties of this dichotomy. Separately, for all threshold functions, we give optimal "biasbased" approximation algorithms generalizing [Chou et al., 2020] while simplifying [Chou et al., 2022]. Finally, we investigate the √ nspace streaming lower bounds in [Chou et al., 2022], and show that they are incomplete for 3AND, i.e., they fail to rule out (α(3AND})ε)approximations in o(√ n) space.more » « less

We generalize Hermite interpolation with error correction, which is the methodology for multiplicity algebraic error correction codes, to Hermite interpolation of a rational function over a field K from function and function derivative values. We present an interpolation algorithm that can locate and correct <= E errors at distinct arguments y in K where at least one of the values or values of a derivative is incorrect. The upper bound E for the number of such y is input. Our algorithm sufficiently oversamples the rational function to guarantee a unique interpolant. We sample (f/g)^(j)(y[i]) for 0 <= j <= L[i], 1 <= i <= n, y[i] distinct, where (f/g)^(j) is the jth derivative of the rational function f/g, f, g in K[x], GCD(f,g)=1, g <= 0, and where N = (L[1]+1)+...+(L[n]+1) >= C + D + 1 + 2(L[1]+1) + ... + 2(L[E]+1) where C is an upper bound for deg(f) and D an upper bound for deg(g), which are input to our algorithm. The arguments y[i] can be poles, which is truly or falsely indicated by a function value infinity with the corresponding L[i]=0. Our results remain valid for fields K of characteristic >= 1 + max L[i]. Our algorithm has the same asymptotic arithmetic complexity as that for classical Hermite interpolation, namely softO(N). For polynomials, that is, g=1, and a uniform derivative profile L[1] = ... = L[n], our algorithm specializes to the univariate multiplicity code decoder that is based on the 1986 WelchBerlekamp algorithm.more » « less

null (Ed.)The approximate degree of a Boolean function f is the least degree of a real polynomial that approximates f pointwise to error at most 1/3. The approximate degree of f is known to be a lower bound on the quantum query complexity of f (Beals et al., FOCS 1998 and J. ACM 2001). We find tight or nearly tight bounds on the approximate degree and quantum query complexities of several basic functions. Specifically, we show the following. kDistinctness: For any constant k, the approximate degree and quantum query complexity of the kdistinctness function is Ω(n3/4−1/(2k)). This is nearly tight for large k, as Belovs (FOCS 2012) has shown that for any constant k, the approximate degree and quantum query complexity of kdistinctness is O(n3/4−1/(2k+2−4)). Image size testing: The approximate degree and quantum query complexity of testing the size of the image of a function [n]→[n] is Ω~(n1/2). This proves a conjecture of Ambainis et al. (SODA 2016), and it implies tight lower bounds on the approximate degree and quantum query complexity of the following natural problems. kJunta testing: A tight Ω~(k1/2) lower bound for kjunta testing, answering the main open question of Ambainis et al. (SODA 2016). Statistical distance from uniform: A tight Ω~(n1/2) lower bound for approximating the statistical distance of a distribution from uniform, answering the main question left open by Bravyi et al. (STACS 2010 and IEEE Trans. Inf. Theory 2011). Shannon entropy: A tight Ω~(n1/2) lower bound for approximating Shannon entropy up to a certain additive constant, answering a question of Li and Wu (2017). Surjectivity: The approximate degree of the surjectivity function is Ω~(n3/4). The best prior lower bound was Ω(n2/3). Our result matches an upper bound of O~(n3/4) due to Sherstov (STOC 2018), which we reprove using different techniques. The quantum query complexity of this function is known to be Θ(n) (Beame and Machmouchi, Quantum Inf. Comput. 2012 and Sherstov, FOCS 2015). Our upper bound for surjectivity introduces new techniques for approximating Boolean functions by lowdegree polynomials. Our lower bounds are proved by significantly refining techniques recently introduced by Bun and Thaler (FOCS 2017).more » « less