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The Kolchin Seminar in Differential Algebra

Hunter College - Room HW706

Saturday February 19th 2005

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Richard Cohn
Rutgers University
A reinterpretation of the difference equation analogue of Ritt's low power condition (Continued)
2:00 - 3:00PM

Abstract: The low power condition introduced by J.F. Ritt is a necessary and sufficient condition for {y} to be a component of {A}, where A is an algebraically irreducible differential polynomial in the polynomial differential ring K{y}. (In its more general form, for any prime differential ideal to be a component). Stated differently, failure of this condition is sufficient but not necessary for A to have a non-zero solution specializing to 0.
There is an analogous condition in difference algebra, the low weight condition. The condition is necessary but not sufficient for {y} to be a component of {B}, where B is an algebraically irreducible polynomial in the difference ring F{y}. Sufficiency holds if B is of second order and in other interesting cases, but fails at third order. Stated differently, failure of the low weight condition is sufficient but not necessary for B to have a non-zero solution specializing to 0.
A full understanding of when {y} is a component of {B} has escaped my investigations and appears difficult. Recently I have found a new interpretation of the low weight condition which sidesteps this difficulty. Failure of the low weight condition is necessary and sufficient for B to have a non-zero solution which specializes "nicely" to 0, where by "nicely" I mean a specialization that extends to a difference closed place, that is a place whose valuation ideal is closed under the difference operator. (The corresponding result is also valid in the differential case). Further development of this idea will probably require Barbara Lando's difference algebra analogue of Morrison theory of differential places.

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Jerry G. Ianni
LaGuardia Community College
On the semigroup structure of synthetic musical scales
3:30-4:30PM

Abstract: A semigroup consisting entirely of idempotents is called a band. We will show that various sets of synthetic musical scales become commutative bands under componentwise ordering of inflections. There is a natural one-to-one correspondence between the commutative bands defined on a nonempty set S and the lower semilattices defined on S. We will present an algorithmic procedure for identifying the ideals of an arbitrary finite commutative band using the Hasse diagram for its associated lower semilattice. We will also give a complete enumeration of the ideals in a few specific cases.
References:
J. Murray Barbour, Synthetic Musical Scales, The American Mathematical Monthly 36 (1929) 155 - 160.
John M. Howie, Fundamentals of Semigroup Theory, Oxford University Press, Oxford, 1995.
Robert M. Mason, Enumeration of Synthetic Musical Scales by Matrix Algebra and a Catalogue of Busoni Scales, Journal of Music Theory 14 (1970) 92 - 126.

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Michael Singer
North Carolina State University
Constructive methods in the inverse problem of differential Galois theory
5:00-6:00PM

Abstract: After a brief review of methods for solving the inverse problem in differential Galois theory, I will present new techniques that allow one to construct differential equations having Galois groups whose connected components are certain semisimple groups.

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Coffee and tea will be served beginning at 1:30 PM
Room HW706

Click here for directions to Hunter College and location of room

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