2,552 research outputs found
Matrix Formula of Differential Resultant for First Order Generic Ordinary Differential Polynomials
In this paper, a matrix representation for the differential resultant of two
generic ordinary differential polynomials and in the differential
indeterminate with order one and arbitrary degree is given. That is, a
non-singular matrix is constructed such that its determinant contains the
differential resultant as a factor. Furthermore, the algebraic sparse resultant
of treated as polynomials in is
shown to be a non-zero multiple of the differential resultant of .
Although very special, this seems to be the first matrix representation for a
class of nonlinear generic differential polynomials
New and Old Results in Resultant Theory
Resultants are getting increasingly important in modern theoretical physics:
they appear whenever one deals with non-linear (polynomial) equations, with
non-quadratic forms or with non-Gaussian integrals. Being a subject of more
than three-hundred-year research, resultants are of course rather well studied:
a lot of explicit formulas, beautiful properties and intriguing relationships
are known in this field. We present a brief overview of these results,
including both recent and already classical. Emphasis is made on explicit
formulas for resultants, which could be practically useful in a future physics
research.Comment: 50 pages, 15 figure
The resultant on compact Riemann surfaces
We introduce a notion of resultant of two meromorphic functions on a compact
Riemann surface and demonstrate its usefulness in several respects. For
example, we exhibit several integral formulas for the resultant, relate it to
potential theory and give explicit formulas for the algebraic dependence
between two meromorphic functions on a compact Riemann surface. As a particular
application, the exponential transform of a quadrature domain in the complex
plane is expressed in terms of the resultant of two meromorphic functions on
the Schottky double of the domain.Comment: 44 page
Berezinians, Exterior Powers and Recurrent Sequences
We study power expansions of the characteristic function of a linear operator
in a -dimensional superspace . We show that traces of exterior
powers of satisfy universal recurrence relations of period .
`Underlying' recurrence relations hold in the Grothendieck ring of
representations of \GL(V). They are expressed by vanishing of certain Hankel
determinants of order in this ring, which generalizes the vanishing of
sufficiently high exterior powers of an ordinary vector space. In particular,
this allows to explicitly express the Berezinian of an operator as a rational
function of traces. We analyze the Cayley--Hamilton identity in a superspace.
Using the geometric meaning of the Berezinian we also give a simple formulation
of the analog of Cramer's rule.Comment: 35 pages. LaTeX 2e. New version: paper substantially reworked and
expanded, new results include
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