Algebraic computation of resolvents without extraneous powers

AbstractThis paper presents an algorithm for computing algebraically relative resolvents which enhances an existing algorithm by avoiding the accumulation of superfluous powers in the intermediate computations. The superfluous power generated at each step is predetermined over a certain quotient ring. As a byproduct, an efficient algorithm for extracting an n-th root of a univariate polynomial is obtained

Algorithms for the universal decomposition algebra

Let k be a field and let f be a polynomial of degree n in k [T]. The symmetric relations are the polynomials in k [X1, ..., Xn] that vanish on all permutations of the roots of f in the algebraic closure of k. These relations form an ideal Is; the universal decomposition algebra is the quotient algebra A := k [X1, ..., Xn]/Is. We show how to obtain efficient algorithms to compute in A. We use a univariate representation of A, i.e. an explicit isomorphism of the form A=k [T]/Q (T), since in this representation, arithmetic operations in A are known to be quasi-optimal. We give details for two related algorithms, to find the isomorphism above, and to compute the characteristic polynomial of any element of A

Exact and approximate polynomial decomposition methods for signal processing applications

Signal processing is a discipline in which functional composition and decomposition can potentially be utilized in a variety of creative ways. From an analysis point of view, further insight can be gained into existing signal processing systems and techniques by reinterpreting them in terms of functional composition. From a synthesis point of view, functional composition offers new algorithms and techniques with modular structure. Moreover, computations can be performed more efficiently and data can be represented more compactly in information systems represented in the context of a compositional structure. Polynomials are ubiquitous in signal processing in the form of z-transforms. In this paper, we summarize the fundamentals of functional composition and decomposition for polynomials from the perspective of exploiting them in signal processing. We compare exact polynomial decomposition algorithms for sequences that are exactly decomposable when expressed as a polynomial, and approximate decomposition algorithms for those that are not exactly decomposable. Furthermore, we identify efficiencies in using exact decomposition techniques in the context of signal processing and introduce a new approximate polynomial decomposition technique based on the use of Structured Total Least Norm (STLN) formulation.Texas Instruments Leadership University Consortium ProgramBose (Firm

An ABC Construction of Number Fields

We describe a general three step method for constructing number fields with Lie-type Galois groups and discriminants factoring into powers of specified primes. The first step involves extremal solutions of the matrix equation ABC = I. The second step involves extremal polynomial solutions of the equation A(x) + B(x) + C(x) = 0. The third step involves integer solutions of the generalized Fermat equation axp + byq + czr = 0. We concentrate here on details associated to the third step and give examples where the field discriminants have the form ±2a3b

Numerical calculation of three-point branched covers of the projective line

We exhibit a numerical method to compute three-point branched covers of the complex projective line. We develop algorithms for working explicitly with Fuchsian triangle groups and their finite index subgroups, and we use these algorithms to compute power series expansions of modular forms on these groups.Comment: 58 pages, 24 figures; referee's comments incorporate

A Fast Algorithm for Computing the Truncated Resultant

International audienceLet P and Q be two polynomials in K[x, y] with degree at most d, where K is a field. Denoting by R ∈ K[x] the resultant of P and Q with respect to y, we present an algorithm to compute R mod x^k in O˜(kd) arithmetic operations in K, where the O˜ notation indicates that we omit polylogarithmic factors. This is an improvement over state-of-the-art algorithms that require to compute R in O˜(d^3) operations before computing its first k coefficients