Let P (x, y) be a rational polynomial. If the curve (P (x, y) = k), k ∈ Q, is irreducible and admits an infinite number of points whose coordinates are integers, Siegel’s theorem implies that the curve is rational. We deal with the case where k is a generic value and prove, in the spirit of the Abhyankar-MohSuzuki theorem, that there exists an algebraic automorphism sending P (x, y) to the poly...

A Bayesian method of estimating an unknown regression curve by a polynomial of random order is proposed. A joint distribution is assigned over both the set of possible orders of the polynomial and the polynomial coeecients. 1995) are used to compute required posteriors. The methodology is extended to polynomials of random order with discontinuties and to piecewise poly-nomials of random order. ...

We prove a generalization of both Pascal’s Theorem and its converse, the Braikenridge–Maclaurin Theorem: If two sets of k lines meet in k2 distinct points, and if dk of those points lie on an irreducible curve C of degree d , then the remaining k(k − d) points lie on a unique curve S of degree k − d. If S is a curve of degree k − d produced in this manner using a curve C of degree d , we say th...

It is well known that the degree-raised Bernstein-Bézier coefficients of degree n of a polynomial g converge to g at the rate 1/n. In this paper we consider the polynomial An(g) of degree ≤ n interpolating the coefficients. We show how An can be viewed as an inverse to the Bernstein operator and that the derivatives An(g) (r) converge uniformly to g at the rate 1/n for all r. We also give an as...

This paper presents a method to selectively elevate the degree of an S-Patch of arbitrary dimension. We consider not only S-Patches with 2D domains but 3D and higher-dimensional domains as well, of which volumetric cage deformations are a subset. We show how to selectively insert control points of a higher degree patch into a lower degree patch while maintaining the polynomial reproduction orde...

We prove a Fourier restriction result for general polynomial curves in Rd . Measuring the Fourier restriction with respect to the affine arclength measure of the curve, we obtain a universal estimate for the class of all polynomial curves of bounded degree. Our method relies on establishing a geometric inequality for general polynomial curves which is of interest in its own right. Applications ...

Let N be a complete, orientable, finite-volume, one-cusped hyperbolic 3-manifold with an ideal triangulation. Using combinatorics of the ideal triangulation of N we construct a plane curve in C×C which contains the squares of eigenvalues of PSL(2,C) representations of the meridian and longitude. We show that the defining polynomial of this curve is related to the PSL(2,C) A-polynomial and has p...

In this paper, we consider the Frobenius endomorphism on twisted Edwards curve and give the characteristic polynomial of the map. Applying the Frobenius endomorphism on twisted Edwards curve, we construct a skew-Frobenius map defined on the quadratic twist of an twisted Edwards curve. Our results show that the Frobenius endomorphism on twisted Edwards curve and the skew-Frobenius endomorphism o...

A polynomial Pythagorean–hodograph (PH) curve r(t) = (x1(t), . . . , xn(t)) in R n is characterized by the property that its derivative components satisfy the Pythagorean condition x′2 1 (t)+ · · ·+x ′2 n (t) = σ 2(t) for some polynomial σ(t), ensuring that the arc length s(t) = ∫ σ(t) dt is simply a polynomial in the curve parameter t. PH curves have thus far been extensively studied in R2 and...