In engineering and applied mathematics, polynomial systems arise whose solution sets contain components of different dimensions and multiplicities. In this article we present algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposition of the solution set. In particular, ignoring multiplicities, our algorithms lay out the decomposition of the set of solutions into irreducible components, by finding, at each dimension, generic points on each component. As by-products, the computation also determines the degree of each component and an upper bound on itsmultiplicity. The bound issharp (i.e., equal to one) for reduced components. The algorithms make essential use of generic projection and interpolation, and can, if desired, describe each irreducible component precisely as the common zeroesof a finite number of polynomials.

Abstract. Let {ar: r ∈ Nd} be a d-dimensional array of numbers for which the generating function F (z): = ∑ r arzr is meromorphic in a neighborhood of the origin. For example, F may be a rational multivariate generating function. We discuss recent results that allow the effective computation of asymptotic expansions for the coefficients of F. Our purpose is to illustrate the use of these techniques on a variety of problems of combinatorial interest. The survey begins by summarizing previous work on the asymptotics of univariate and multivariate generating functions. Next we describe the Morse-theoretic underpinnings of some new asymptotic techniques. We then quote and summarize these results in such a way that only elementary analyses are needed to check hypotheses and carry out computations. The remainder of the survey focuses on combinatorial applications, such as enumeration of words with forbidden substrings, edges and cycles in graphs, polyominoes, and descents in permutations. After the individual examples, we discuss three broad classes of examples, namely, functions derived via the transfer matrix method, those derived via the kernel method, and those derived via the method of Lagrange inversion. These methods have

The paper [FGS] uses the classification of finite simple groups and covering theory in positive characteristic to solve Carlitz's conjecture (1966). We consider only separable polynomials; their derivative is nonzero. Then, f # Fq [x] is exceprional if it acts as a permutation map on infinitely many finite extensions of the finite field Fq , q = p a for some prime p. Carlitz's conjecture says f must be of odd degree (if p is odd). The main theorem of [FGS; Theorem 14.1] restricts the list of possible geometric monodromy groups of exceptional indecomposable polynomials (§1.1): either p = 2 or 3 or these must be affine groups. The proof of Carlitz's conjecture motivates considering general exceptional covers of nonsingular projective algebraic curves. For historical reasons we sometimes call these Schur covers [Fr2]. Suppose # : X # P 1 is an exceptional cover over Fq . Then, for some integer s, there is au/I--L x x x # X(F q t ) over each z # P 1 (F q t ) foreac h integer t with (t, s) = 1. In particular /5/ |X(F q t )| = q t +1 when (t, s) = 1. We include a complete proof that exceptionality is equivalent to a statement about the geometric/arithmetic monodromy pair of the cover. Theorem 2.5 shows all geometric/arithmetic monodromy pairs satisfying necessary conditions (§1.1-§1.2) derive from covers over Fp for all suitably large primes p. Other topics: (i) How modular curve points over finite fields explicitly produce rational function exceptional covers of prime degree (Corollary 3.5). (ii) How fiber products produce abundant general exceptional covers (Lemma 3.7). (iii) How Müller-Chen-Matthews produced exceptional polynomials with nonsolvable monodromy group ($1.7). (iv) How general exceptional covers realize curves of high genus over Fq with q small and |X(F q t # )| large for...

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this paper we consider RCT over an arbitrary zerocharacteristic field F with branching signs f=; 6=g and also more customary RCT over reals with branching signs f; ?g. We remind (see e.g. [24], [19], [13]) that RCT T = fT ff g ff is a collection of computation trees T ff which are chosen with the probabilities p ff 0;

The solution set V of a polynomial system, i.e., the set of common zeroes of a set of multivariate polynomials with complex coefficients, may contain several components, e.g., points, curves, surfaces, etc. Each component has attached to it a number of quantities, one of which is its dimension. Given a numerical approximation to a point p on the set V, this article presents an efficient algorithm to compute the maximum dimension of the irreducible components of V which pass through p, i.e., a local dimension test. Such a test is a crucial element in the homotopy-based numerical irreducible decomposition algorithms of Sommese, Verschelde, and Wampler. This article presents computational evidence to illustrate that the use of this new algorithm greatly reduces the cost of so-called “junk-point filtering, ” previously a significant bottleneck in the computation of a numerical irreducible decomposition. For moderate size examples, this results in well over an order of magnitude improvement in the computation of a numerical irreducible decomposition. As the computation of a numerical irreducible decomposition is a fundamental backbone operation, gains in efficiency in the irreducible decomposition algorithm carry over to the many computations which require this decomposition as an initial step. Another feature of a local dimension test is that one can now compute the irreducible components in a prescribed dimension without first computing the numerical irreducible decomposition of all higher dimensions. For example, one may compute the isolated solutions of a polynomial system without having to carry out the full numerical irreducible decomposition.

A stratification of a set, e.g. an analytic variety, is, roughly, a partition of it into manifolds so that these manifolds fit together “regularly”. Stratification theory was originated by Thom and Whitney for algebraic and analytic sets. It was one of the key ingredients in Mather’s proof of the topological stability theorem [Ma] (see [GM] and [PW] for the

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this paper.