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

Abstract

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...

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

“…Here there are 20 contributing singularities in multiple orthants, many flats have contributing singularities in multiple quadrants, and several contributing points have irrational coordinates. We use the symbolic-numeric methods of Melczer and Salvy [25] to store the coordinates of the contributing points: for each contributing point σ the algorithm outputs an algebraic number α defined by a square-free integer polynomial P (u) and isolating interval, together with integer polynomials Q j (u) for each coordinate, such that the jth coordinate σ j = Q j (α)/P (α).…”

“…There is some randomness in selecting the polynomial P (u); see Melczer and Salvy [25] for the advantages of this representation and how to compute it, and the Maple worksheet accompanying this paper for more details on this example. Computing the asymptotic contributions of each contributing point here gives dominant asymptotics of the form…”

“…When the denominator under consideration is a product of linear factors, Bertozzi and McKenna [8] determine this set for a few examples and speculated on the existence of a theory to compute the set in general. Outside of the linear denominator case, which we cover extensively in this paper, when the singular set of F is a manifold then it is known how to compute the contributing points both in the bivariate case [12] and when the contributing points lie on the boundary of the domain of convergence of the power series [25]. There is currently no known algorithm to determine contributing singularities for general meromorphic (or even rational) functions, and such an algorithm would need to decide certain deep topological questions.…”

Multivariate Analytic Combinatorics for Cost Constrained Channels and Subsequence Enumeration

“…In order to apply the standard results of ACSV, strong information about the singular set is needed. In particular, we must identify minimal critical points and points with the same coordinate-wise modulii; these are real algebraic properties (involving relationships among the moduli of complex variables) and thus expensive to determine computationally even on many explicit bivariate examples (see [MS21] for a discussion of the complexity of such operations). These issues are also complicated by the non-smoothness of the singular set of the generating functions under consideration.…”

Absolute Root Separation

“…We proved a more precise version of the first bound in a previous work [2], where we showed that the exponent of H(P ) is optimal in that case. More detailed but less precise bounds for the second case can be found in [10,Lemma 3.6] and [11,Lemma 53]. The proof of Theorem 1 is based on constructing auxiliary polynomials with integer coefficients of controlled height whose roots contain the desired difference.…”