Georgia Institute of Technology College of Sciences

Let I be an ideal in a polynomial ring R := F[x_n,...,x_1] Let A := R/I be the corresponding quotient ring, and let Q(A) be its field of fractions. The integral closure C(A, Q(A)) of A in Q(A) is a subring of the latter. But it is often given as a separate quotient ring, a presentation.Surprisingly, different computer algebra systems (Magma, Macaulay2, and Singular) choose to produce very different presentations. Some of these opt for presentations that have seductive forms, but miss the most important, namely a form that allows for determining when elements of Q(A) are in C(A,Q(A)). This is called membership and is directly related to determining isomorphism.

For two polynomials G(X), H(Y) with rational coefficients, when does G(X) = H(Y) have infinitely many solutions over the rationals? Such G and H have been classified in various special cases by previous mathematicians. A theorem of Faltings (the Mordell conjecture) states that we need only analyze curves with genus at most 1.In my thesis (and more recent work), I classify G(X) = H(Y) defining irreducible genus zero curves. In this talk I'll present the infinite families which arise in this classification, and discuss the techniques used to complete the classification.I will also discuss in some detail the examples of polynomial which occur in the classification. The most interesting infinite family of polynomials are those H(Y) solving a Pell Equation H(Y)^2 - P(Y)Q(Y)^2 = 1. It turns out to be difficult to describe these polynomials more explicitly, and yet we can completely analyze their decompositions, how many such polynomials there are of a fixed degree, which of them are defined over the rationals (as opposed to a larger field), and other properties.

Granville and Soundararajan have recently introduced thenotion of pretentiousness in the study of multiplicative functions ofmodulus bounded by 1, essentially the idea that two functions whichare similar in a precise sense should exhibit similar behavior. Itturns out, somewhat surprisingly, that this does not directly extendto detecting power cancellation - there are multiplicative functionswhich exhibit as much cancellation as possible in their partial sumsthat, modified slightly, give rise to functions which exhibit almostas little as possible. We develop two new notions of pretentiousnessunder which power cancellation can be detected, one of which appliesto a much broader class of multiplicative functions. This work isjoint with Junehyuk Jung.

Geometric modeling builds computer models for industrial design and manufacture from basic units, called patches, such as, Bézier curves and surfaces. The control polygon of a Bézier curve is well-defined and has geometric significance—there is a sequence of weights under which the limiting position of the curve is the control polygon. For a Bezier surface patch, there are many possible polyhedral control structures, and none are canonical. In this talk, I will present a not necessarily polyhedral control structure for surface patches, regular control surfaces, which are certain C^0 spline surfaces. While not unique, regular control surfaces are exactly the possible limiting positions of a Bezier patch when the weights are allowed to vary. While our primary interest is to explain the meaning of control nets for the classical rational Bezier patches, we work in the generality of Krasauskas’ toric Bezier patches. Toric Bezier patches are multi-sided parametric patches based on the geometry of toric varieties and depend on a polytope and some weights. Our results rely upon a construction in computational algebraic geometry called a toric degeneration.