**The Kolchin Seminar in Differential
Algebra **

**Saturday February 19 ^{th}
2005**

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**Richard Cohn**Rutgers
University

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

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**