arXiv subjects

Seth Dutter

Publications and source records attributed to Seth Dutter.

5 recordsLinked to original sources

arXiv2026

Visualization of Complex Projective Curves

We introduce a nonlinear map $\alpha:\mathbb{C}^2\rightarrow\mathbb{R}^3$ with the purpose of visualizing curves. Basic properties of $\alpha$ are proved, including preservation of orthogonality, recovery of the magnitudes of vectors in the preimage, and continuous extension of $\alpha$ to $\widetilde{\alpha}:\mathbb{P}^2_\mathbb{C}\rightarrow\mathbb{R}^3$. For plane curves $Z\subset\mathbb{P}^2_\mathbb{C}$, it is proved that $\widetilde{\alpha}(Z)$ is the union of boundaries of star-shaped domains. Methods are established to descend finite-order automorphisms of smooth projective curves to rotations of their images in $\mathbb{R}^3$. Efficient techniques for creating meshes and ray-traced images of $\widetilde{\alpha}(Z)$ are developed.

math.CV↗

arXiv2016

Waring's Problem for Polynomial Rings and the Digit Sum of Exponents

Let $F$ be an algebraically closed field of characteristic $p>0$. In this paper we develop methods to represent arbitrary elements of $F[t]$ as sums of perfect $k$-th powers for any $k\in\mathbb{N}$ relatively prime to $p$. Using these methods we establish bounds on the necessary number of $k$-th powers in terms of the sum of the digits of $k$ in its base-$p$ expansion. As one particular application we prove that for any fixed prime $p>2$ and any $\epsilon>0$ the number of $(p^r-1)$-th powers required is $\mathcal{O}\left(r^{(2+\epsilon)\ln(p)}\right)$ as a function of $r$.

math.NT↗

arXiv2015

Reversals of Least-Squares Estimates and Model-Independent Estimation for Directions of Unique Effects

When a linear model is adjusted to control for additional explanatory variables the sign of a fitted coefficient may reverse. Here these reversals are studied using coefficients of determination. The resulting theory can be used to determine directions of unique effects in the presence of substantial model uncertainty. This process is called model-independent estimation when the estimates are invariant across changes to the model structure. When a single covariate is added, the reversal region can be understood geometrically as an elliptical cone of two nappes with an axis of symmetry relating to a best-possible condition for a reversal using a single coefficient of determination. When a set of covariates are added to a model with a single explanatory variable, model-independent estimation can be implemented using subject matter knowledge. More general theory with partial coefficients is applicable to analysis of large data sets. Applications are demonstrated with dietary health data from the United Nations. Necessary conditions for Simpson's paradox are derived.

stat.ME↗

arXiv2014

Gaps in Taylor series of algebraic functions

Let $f$ be a rational function on an algebraic curve over the complex numbers. For a point $p$ and local parameter $x$ we can consider the Taylor series for $f$ in the variable $x$. In this paper we give an upper bound on the frequency with which the terms in the Taylor series have $0$ as their coefficient.

math.AG↗

arXiv2010

Logarithmic Jet Spaces and Intersection Multiplicities

The theory of relative logarithmic jet spaces is developed for log schemes. With this theory the existence of bounds of intersection multiplicities of curves and divisors on certain log schemes is established. This result extends those of Noguchi and Winkelmann by replacing the semi-abelian condition by a differential one.

math.AG↗