Uniqueness of certain polynomials constant on a line

We study a question with connections to linear algebra, real algebraic geometry, combinatorics, and complex analysis. Let $p(x,y)$ be a polynomial of degree $d$ with $N$ positive coefficients and no negative coefficients, such that $p=1$ when $x+y=1$. A sharp estimate $d \leq 2N-3$ is known. In this paper we study the $p$ for which equality holds. We prove some new results about the form of these "sharp" polynomials. Using these new results and using two independent computational methods we give a complete classification of these polynomials up to $d=17$. The question is motivated by the problem of classification of CR maps between spheres in different dimensions.Comment: 20 pages, latex; removed section 10 and address referee suggestions; accepted to Linear Algebra and its Application

Invariant proper holomorphic maps between balls

We consider proper holomorphic maps between balls that are invariant under the action of finite groups of unitary matrices. We are primarily interested in actions of groups that are fixed-point-free; for purposes of comparison we will briefly consider matrix groups that act with fixed points (that is, groups that have at least one nontrivial element with an eigenvalue of one) in the last chapter. Forstneric showed that given any finite unitary fixed-point-free matrix group, there exists a proper holomorphic map from the ball in the appropriate dimensional complex Euclidean space to a higher dimensional ball, that is invariant under the action of that group. He showed on the other hand that if we also require the map to be smooth to the boundary, then many groups are ruled out.One of our main results is the following theorem: if f is a proper holomorphic map between balls that is invariant under the action of some finite fixed-point-free matrix subgroup of a unitary group (acting on the domain of f), and, in addition, smooth to the boundary, then necessarily that group is cyclic. We rule out some of these cyclic unitary groups as well. We give corollaries concerning the nonexistence of smooth CR mappings from certain spherical space forms to spheres.We next prove some propositions related to the theory of polynomial proper mappings between balls. As another important result, in cases where there are known finite fixed-point-free matrix group-invariant mappings we classify all such maps in terms of a group-basic map. In a subsequent chapter we investigate existence and nonexistence of various sorts of polynomial proper maps between balls, mostly invariant under some matrix group action, from a combinatorial perspective. We give a simple means of depicting monomial mappings from the ball in two-dimensional space, and show some applications. As a final theorem, we show how proper holomorphic maps between balls, invariant under the action of finite matrix groups possibly acting with fixed points, can be "constructed". This uses a technique developed by Low. We derive some interesting examples from this construction

Cylinders through five points: complex and real enumerative geometry

Abstract. It is known that five points in R 3 generically determine a finite number of cylinders containing those points. We discuss ways in which it can be shown that the generic (complex) number of solutions, with multiplicity, is six, of which an even number will be real valued and hence correspond to actual cylinders in R 3. We partially classify the case of no real solutions in terms of the geometry of the five given points. We also investigate the special case where the five given points are coplanar, as it differs from the generic case for both complex and real valued solution cardinalities

Rolling a coin around a coin

Curves, Plane GeometryThe red circle rolls on the outside of the black circle of radius 1. (The blue point shows how it rolls.) An invisible point at distance q from the center of the black circle is attached to the red circle, tracing out a path as the red circle rolls; such curves are called epitrochoids. When the circles have the same radius, you can see that the blue point is oriented in the same direction after half a revolution. Therefore a coin rolling completely around another coin of the same size goes through two complete revolutionsComponente Curricular::Educação Superior::Ciências Exatas e da Terra::Matemátic

Rolling a coin around a coin

Curves, Plane GeometryThe red circle rolls on the outside of the black circle of radius 1. (The blue point shows how it rolls.) An invisible point at distance q from the center of the black circle is attached to the red circle, tracing out a path as the red circle rolls; such curves are called epitrochoids. When the circles have the same radius, you can see that the blue point is oriented in the same direction after half a revolution. Therefore a coin rolling completely around another coin of the same size goes through two complete revolutionsComponente Curricular::Educação Superior::Ciências Exatas e da Terra::Matemátic

Half-GCD and Fast . . .

Over the past few decades several variations on a “half GCD ” algorithm for obtaining the pair of terms in the middle of a Euclidean sequence have been proposed. In the integer case algorithm design and proof of correctness are complicated by the effect of carries. This paper will demonstrate a variant with a relatively simple proof of correctness. We then apply this to the task of rational recovery for a linear algebra solver