We exhibit a condition-based analysis of the adaptive subdivision algorithm due to Plantinga and Vegter. e first complexity analysis of the PV Algorithm is due to Burr, Gao and Tsigaridas who proved a O 2 τ d 4 log d worst-case cost bound for degree d plane curves with maximum coefficient bit-size τ .is exponential bound, it was observed, is in stark contrast with the good performance of the algorithm in practice. More in line with this performance, we show that, with respect to a broad family of measures, the expected time complexity of the PV Algorithm is bounded by O(d 7 ) for real, degree d, plane curves. We also exhibit a smoothed analysis of the PV Algorithm that yields similar complexity estimates. To obtain these results we combine robust probabilistic techniques coming from geometric functional analysis with condition numbers and the continuous amortization paradigm introduced by Burr, Krahmer and Yap. We hope this will motivate a fruitful exchange of ideas between the different approaches to numerical computation. CCS CONCEPTS•Mathematics of computing → Computations on polynomials; Interval arithmetic; • eory of computation → Design and analysis of algorithms; Computational geometry; KEYWORDS computational algebraic geometry, numerical methods, adaptive subdivision methods, isotopy of curves, complexity ACKNOWLEDGMENTSWe thank Michael Burr for useful discussions.

Generalising prior work on the rank of random matrices over finite fields [Coja-Oghlan and Gao 2018], we determine the rank of a random matrix A with prescribed numbers of non-zero entries in each row and column over any field F. The rank formula turns out to be independent of both the field and the distribution of the non-zero matrix entries. The proofs are based on a blend of algebraic and probabilistic methods inspired by ideas from mathematical physics.MSC: 05C80, 60B20, 94B05Amin Coja-Oghlan's research received support under DFG CO 646/3. Alperen A. Ergür partially funded by Einstein Foundation, Berlin.

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We consider the sensitivity of real zeros of polynomial systems with respect to perturbation of the coefficients, and extend our earlier probabilistic estimates for the condition number in two directions: (1) We give refined bounds for the condition number of random structured polynomial systems, depending on a variant of sparsity and an intrinsic geometric quantity called dispersion.(2) Given any structured polynomial system P , we prove the existence of a nearby well-conditioned structured polynomial system Q, with explicit quantitative estimates.Our underlying notion of structure is to consider a linear subspace E i of the space H di of homogeneous n-variate polynomials of degree d i , let our polynomial system P be an element of E := E 1 × · · · × E n−1 , and let dim(E) := dim(E 1 ) + · · · + dim(E n−1 ) be our measure of sparsity. The dispersion σ(E) provides a rough measure of how suitable the tuple E is for numerical solving.Part I of this series studied how to extend probabilistic estimates of a condition number defined by Cucker to a family of measures going beyond the weighted Gaussians often considered in the current literature. We continue at this level of generality, using tools from geometric functional analysis.

Consider a system f 1 (x) = 0, . . . , fn(x) = 0 of n random real polynomial equations in n variables, where each f i has a prescribed set of exponent vectors described by a set A ⊆ N n of cardinality t. Assuming that the coefficients of the f i are independent Gaussians of any variance, we prove that the expected number of zeros of the random system in the positive orthant is bounded from above by 1 2 n−1 t n .

We study the complexity of approximating complex zero sets of certain nvariate exponential sums. We show that the real part, R, of such a zero set can be approximated by the (n − 1)-dimensional skeleton, T , of a polyhedral subdivision of R n . In particular, we give an explicit upper bound on the Hausdorff distance: ∆(R, T ) = O t 3.5 /δ , where t and δ are respectively the number of terms and the minimal spacing of the frequencies of g. On the side of computational complexity, we show that even the n = 2 case of the membership problem for R is undecidable in the Blum-Shub-Smale model over R, whereas membership and distance queries for our polyhedral approximation T can be decided in polynomial-time for any fixed n.

Isolating the real roots of univariate polynomials is a fundamental problem in symbolic computation and it is arguably one of the most important problems in computational mathematics. The problem has a long history decorated with numerous ingenious algorithms and furnishes an active area of research. However, the worst-case analysis of root-finding algorithms does not correlate with their practical performance. We develop a smoothed analysis framework for polynomials with integer coefficients to bridge the gap between the complexity estimates and the practical performance. In this setting, we derive that the expected bit complexity of Descartes solver to isolate the real roots of a polynomial, with coefficients uniformly distributed, is O B (d 2 + dτ ), where d is the degree of the polynomial and τ the bitsize of the coefficients.