Stationary Points at Infinity for Analytic Combinatorics

Baryshnikov, Yuliy; Melczer, Stephen; Pemantle, Robin

doi:10.1007/s10208-021-09523-x

Cited by 7 publications

(11 citation statements)

References 40 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If a < h min and C is a cycle in M ≤a , then C can be continuously deformed in M so that max{h(z) : z ∈ C} is as small as desired. This follows from a non-proper extension of the fundamental Morse lemma described in [4] (because there are no so-called critical points at infinity in our case), or more directly from a concrete modification of the downward gradient flow on M, using a partition of unity to keep the flow away from all strata of V. It follows that the pairs H d (M, M ≤a ) with a < h min are naturally isomorphic. We denote any of these naturally equivalent pairs by (M, −∞) and use the symbol…”

Section: Topological Definitionsmentioning

confidence: 90%

“…, z d ) k , leading to a central limit theorem and Gaussian behaviour of the generating function coefficients. Since the early 2000s, a systematic program to determine asymptotics of multivariate generating function coefficients has been developed by the analytic combinatorics in several variables project 1 [32,33,5,34,3,24]. In the earliest of these works [32,33] the deformations were accomplished by ad hoc surgeries.…”

Section: Acsv Backgroundmentioning

confidence: 99%

“…In the earliest of these works [32,33] the deformations were accomplished by ad hoc surgeries. In [5], existence of the relevant deformations was shown to follow from the theory of hyperbolic functions [1] and a Morse-theoretic structure was provided to compute and interpret them; this Morse-theoretic approach was recently made rigorous in [3] by working around certain standard assumptions of Morse theory that don't hold in ACSV contexts.…”

Section: Acsv Backgroundmentioning

confidence: 99%

“…where 1 t denotes the t-dimensional all ones vector. Since the a j in (18) are positive, replacing the domain of integration in (19) by any of the 2 s − 1 imaginary fibers with basepoints (±1 s , 1 t−s , 0) not equal to (1 t , 0) results in an integral of exponentially smaller growth 4 . Taking a signed sum of these integrals thus introduces an exponentially negligible error and results in a (d − s)-dimensional integral obtained by taking univariate residues in the variables w 1 , .…”

Section: Note Thatmentioning

confidence: 99%

“…(x, y) 2 (x, y) 5 (x, y)4 + 1 4 1 (x, y) 3 (x, y) 5 (x, y) 4 .Each of these summands is now simple, and we may use our Maple implementation of the results in Section 4 to find asymptotics in direction r = (1, 2, 3), say. The summands have dominant coefficient asymptotics that [x n y 2n z 3n ]F (x, y, z) = −…”

mentioning

confidence: 99%

See 4 more Smart Citations

Asymptotics of multivariate sequences IV: generating functions with poles on a hyperplane arrangement

Baryshnikov¹,

Melczer²,

Pemantle³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

Let F (z1, . . . , z d ) be the quotient of an analytic function with a product of linear functions. Working in the framework of analytic combinatorics in several variables, we compute asymptotic formulae for the Taylor coefficients of F using multivariate residues and saddle-point approximations. Because the singular set of F is the union of hyperplanes, we are able to make explicit the topological decompositions which arise in the multivariate singularity analysis. In addition to effective and explicit asymptotic results, we provide the first results on transitions between different asymptotic regimes, and provide the first software package to verify and compute asymptotics in non-smooth cases of analytic combinatorics in several variables. It is also our hope that this paper will serve as an entry to the more advanced corners of analytic combinatorics in several variables for combinatorialists.

show abstract

Section: Topological Definitionsmentioning

confidence: 90%

Section: Acsv Backgroundmentioning

confidence: 99%

Section: Acsv Backgroundmentioning

confidence: 99%

Section: Note Thatmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Asymptotics of multivariate sequences IV: generating functions with poles on a hyperplane arrangement

Baryshnikov¹,

Melczer²,

Pemantle³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Asymptotics of Multivariate Sequences IV: Generating Functions with Poles on a Hyperplane Arrangement

2023

View full text Add to dashboard Cite

Effective coefficient asymptotics of multivariate rational functions via semi-numerical algorithms for polynomial systems

Melczer

Salvy

2021

Journal of Symbolic Computation

Self Cite

View full text Add to dashboard Cite

The coefficient sequences of multivariate rational functions appear in many areas of combinatorics. Their diagonal coefficient sequences enjoy nice arithmetic and asymptotic properties, and the field of analytic combinatorics in several variables (ACSV) makes it possible to compute asymptotic expansions. We consider these methods from the point of view of effectivity. In particular, given a rational function, ACSV requires one to determine a (generically) finite collection of points that are called critical and minimal. Criticality is an algebraic condition, meaning it is well treated by classical methods in computer algebra, while minimality is a semi-algebraic condition describing points on the boundary of the domain of convergence of a multivariate power series. We show how to obtain dominant asymptotics for the diagonal coefficient sequence of multivariate rational functions under some genericity assumptions using symbolic-numeric techniques. To our knowledge, this is the first completely automatic treatment and complexity analysis for the asymptotic enumeration of rational functions in an arbitrary number of variables.We also propose an algorithm finding minimal critical points in many cases, even when combinatoriality is not assumed, at the price of an increase in complexity. Our result in that case is the following, which is stated precisely in Theorem 57 below.Result 2. Let F (z) ∈ Z(z 1 , . . . , z n ) be a rational function with numerator and denominator of degrees at most d and coefficients of absolute value at most 2 h . Assuming that F satisfies certain verifiable assumptions stated in Section 3.3, then F admits a finite number of minimal critical points that can be determined iñ O hd 9n+5 2 3n bit operations. From there, the asymptotics of the diagonal coefficients follow with the same complexity as in Result 1.Aside from the existence of minimal critical points, we conjecture that the assumptions on F required to apply Theorem 57 in the non-combinatorial case hold generically. dimensional.We start with the extended system, the other one is similar. This system has n + 2 equations of degree at most d in n+2 variables. Example 36 shows that it is evaluated by a straight-line program of length O(L+n). We partition the variables into three blocks z, λ, t.The bound C n (d) is obtained as the sum of the non-zero coefficients of θ z , θ λ , θ t inleading to C n (d) = nd n+1 . The computation for the height is similar. With η 1 = h + log(n + 1)d, η 2 = · · · = η n+1 = h + d + log(n + 1)d + 1, η n+2 = h + 2 log(n + 1)dandit follows that H n (η, d) = O(Dn(d + n)(h + log(n + 1)d)) =Õ(dD(h + d)). The complexity then follows from injecting these quantities in the previous proposition. The bounds for the system formed by the critical point equations only are derived as above with simpler computationsleading to nD for the degree andÕ (D(h + d)) for the height.Example 44. These inequalities reflect a perceptible growth of the sizes with the number of variables in computations.Starting from the same system as in Ex...

show abstract

Stationary Points at Infinity for Analytic Combinatorics

Cited by 7 publications

References 40 publications

Asymptotics of multivariate sequences IV: generating functions with poles on a hyperplane arrangement

Asymptotics of multivariate sequences IV: generating functions with poles on a hyperplane arrangement

Asymptotics of Multivariate Sequences IV: Generating Functions with Poles on a Hyperplane Arrangement

Effective coefficient asymptotics of multivariate rational functions via semi-numerical algorithms for polynomial systems

Contact Info

Product

Resources

About