679 research outputs found
Formalization of a Newton Series Representation of Polynomials
International audienceWe formalize an algorithm to change the representation of a polynomial to a Newton power series. This provides a way to compute efficiently polynomials whose roots are the sums or products of roots of other polynomials, and hence provides a base component of efficient computation for algebraic numbers. In order to achieve this, we formalize a notion of truncated power series and develop an abstract theory of poles of fractions
Analysis of Reaction Network Systems Using Tropical Geometry
We discuss a novel analysis method for reaction network systems with
polynomial or rational rate functions. This method is based on computing
tropical equilibrations defined by the equality of at least two dominant
monomials of opposite signs in the differential equations of each dynamic
variable. In algebraic geometry, the tropical equilibration problem is
tantamount to finding tropical prevarieties, that are finite intersections of
tropical hypersurfaces. Tropical equilibrations with the same set of dominant
monomials define a branch or equivalence class. Minimal branches are
particularly interesting as they describe the simplest states of the reaction
network. We provide a method to compute the number of minimal branches and to
find representative tropical equilibrations for each branch.Comment: Proceedings Computer Algebra in Scientific Computing CASC 201
A verified Common Lisp implementation of Buchberger's algorithm in ACL2
In this article, we present the formal verification of a Common
Lisp implementation of Buchberger's algorithm for computing
Gröbner bases of polynomial ideals. This work is carried out in
ACL2, a system which provides an integrated environment where
programming (in a pure functional subset of Common Lisp) and
formal verification of programs, with the assistance of a theorem
prover, are possible. Our implementation is written in a real
programming language and it is directly executable within the
ACL2 system or any compliant Common Lisp system. We provide
here snippets of real verified code, discuss the formalization details
in depth, and present quantitative data about the proof effort
Root finding with threshold circuits
We show that for any constant d, complex roots of degree d univariate
rational (or Gaussian rational) polynomials---given by a list of coefficients
in binary---can be computed to a given accuracy by a uniform TC^0 algorithm (a
uniform family of constant-depth polynomial-size threshold circuits). The basic
idea is to compute the inverse function of the polynomial by a power series. We
also discuss an application to the theory VTC^0 of bounded arithmetic.Comment: 19 pages, 1 figur
Enumerative Real Algebraic Geometry
Enumerative Geometry is concerned with the number of solutions to a
structured system of polynomial equations, when the structure comes from
geometry. Enumerative real algebraic geometry studies real solutions to such
systems, particularly a priori information on their number. Recent results in
this area have, often as not, uncovered new and unexpected phenomena, and it is
far from clear what to expect in general. Nevertheless, some themes are
emerging.
This comprehensive article describe the current state of knowledge,
indicating these themes, and suggests lines of future research. In particular,
it compares the state of knowledge in Enumerative Real Algebraic Geometry with
what is known about real solutions to systems of sparse polynomials.Comment: Revised, corrected version. 40 pages, 18 color .eps figures. Expanded
web-based version at http://www.math.umass.edu/~sottile/pages/ERAG/index.htm
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