Searcharxiv⌕ Search

arXiv subjects

Piotr Migus

Publications and source records attributed to Piotr Migus.

9 recordsLinked to original sources
arXiv2026

Generic degrees of real polynomial Keller maps with non-dense image

We determine the possible generic degrees of real polynomial Keller maps with non-dense image, where density is understood in the Euclidean topology. For every dimension $n\geq3$, these degrees are exactly the even integers $d\geq4$. They are realized in dimension three by an explicit family $G_d\colon\mathbb{R}^{3}\to\mathbb{R}^{3}$, and hence in every higher dimension by stabilization. In dimension two, any such degree is an even integer at least six, and no such maps exist if the planar Jacobian conjecture holds; in dimension one, none exist. For the family $G_d$, we describe the image exactly and show that no $G_d$ omits a half-space. We also show that the maximal cardinality of a real fibre is not determined by the generic degree and is not uniformly bounded in this class.

math.AG↗
arXiv2026

Bi-Lipschitz invariance of Newton polygons along gradient canyons

We study bi-Lipschitz right-equivalence of holomorphic function germs $f:(\mathbb{C}^2,0)\to(\mathbb{C},0)$ via polar arcs and gradient canyons. For a polar arc $\gamma$ we consider the Newton polygon of $f_x(X+\gamma(Y),Y)$ and define its augmentation by adjoining the point $(0,\operatorname{ord} f(\gamma(y),y)-1)$. We prove that the resulting augmented Newton polygon is constant along each gradient canyon of degree $>1$ and is invariant under bi-Lipschitz right-equivalence. Moreover, its compact edges decompose into a topological part and a Lipschitz part: the latter encodes, through simple intercept relations, the second-level Henry-Parusi\'nski type invariants. As applications, we obtain two numerical bi-Lipschitz invariants attached to a canyon: its polar multiplicity and, via the Koike-Kuo-P\u{a}unescu curvature formula, the total asymptotic Gaussian curvature concentrated in it.

math.CV↗
arXiv2025

Newton-Puiseux Analysis for Interpretability and Calibration of Complex-Valued Neural Networks

Complex-valued neural networks (CVNNs) are particularly suitable for handling phase-sensitive signals, including electrocardiography (ECG), radar/sonar, and wireless in-phase/quadrature (I/Q) streams. Nevertheless, their \emph{interpretability} and \emph{probability calibration} remain insufficiently investigated. In this work, we present a Newton--Puiseux framework that examines the \emph{local decision geometry} of a trained CVNN by (i) fitting a small, kink-aware polynomial surrogate to the \emph{logit difference} in the vicinity of uncertain inputs, and (ii) factorizing this surrogate using Newton--Puiseux expansions to derive analytic branch descriptors, including exponents, multiplicities, and orientations. These descriptors provide phase-aligned directions that induce class flips in the original network and allow for a straightforward, \emph{multiplicity-guided} temperature adjustment for improved calibration. We outline assumptions and diagnostic measures under which the surrogate proves informative and characterize potential failure modes arising from piecewise-holomorphic activations (e.g., modReLU). Our phase-aware analysis identifies sensitive directions and enhances Expected Calibration Error in two case studies beyond a controlled $\C^2$ synthetic benchmark -- namely, the MIT--BIH arrhythmia (ECG) dataset and RadioML 2016.10a (wireless modulation) -- when compared to uncalibrated softmax and standard post-hoc baselines. We also present confidence intervals, non-parametric tests, and quantify sensitivity to inaccuracies in estimating branch multiplicity. Crucially, this method requires no modifications to the architecture and applies to any CVNN with complex logits transformed to real moduli.

cs.LG↗
arXiv2021

Clustering polar curves

This essay builds on the idea of grouping the polar curves of 2-variable function germs into polar clusters. In the topological category, one obtains a bijective correspondence between certain partitions of the polar quotients of two topologically equivalent function germs. We explain how this bijective correspondence may be refined in the Lipschitz category in terms of the associated gradient canyons.

math.CV↗
arXiv2015

Local $C^r$-right equivalence of $C^{r+1}$ functions

Let $f,g:(\mathbb{R}^n,0)\rightarrow (\mathbb{R},0)$ be $C^{r+1}$ functions, $r\in \mathbb{N}$. We will show that if $\nabla f(0)=0$ and there exist a neigbourhood $U$ of $0\in \mathbb{R}^n$ and a constant $C>0$ such that $$ \left|\partial^m(g-f)(x)\right|\leq C \left|\nabla f(x)\right|^{r+2-|m|}, \quad x\in U, $$ for any $m\in \mathbb{N}_0^n$ such that $|m|\leq r$, then there exists a $C^r$ diffeomorphism $φ:(\mathbb{R}^n,0)\rightarrow (\mathbb{R}^n,0)$ such that $f=g\circ φ$ in a neighbourhood of $0$.

math.AG↗
arXiv2014

$C^r$-right equivalence of analytic functions

Let $f,g:(\mathbb{R}^n,0)\rightarrow (\mathbb{R},0)$ be analytic functions. We will show that if $\nabla f(0)=0$ and $g-f \in (f)^{r+2}$ then $f$ and $g$ are $C^r$-right equivalent, where $(f)$ denote ideal generated by $f$ and $r\in \mathbb{N}$.

math.AG↗