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Sufficient conditions for the surjectivity of radical curve parametrizations

Jorce Caravantes, J. Rafael Sendra, David Sevilla, Carlos Villarino

TL;DR

This paper introduces a notion of surjectivity for radical curve parametrizations and provides sufficient conditions to guarantee that a radical parametrization covers the entire radical curve. By extending techniques from rational parametrizations, it uses a lifting strategy based on the Extension Theorem to propagate a point from the projection of the radical data to a full preimage, and introduces the notions of guilty and suspicious polynomials to manage degree losses caused by radical expressions. The main result identifies a concrete, checkable criterion: if there exists an index $i$ with $p_i$ not guilty and $\deg p_i>\deg q_i$, and for all $i$ the ideals $I_i(\mathcal{P})$ equal $\mathbb{C}[t,\overline{\Delta}]$, then the parametrization is surjective; variations yield practical, computable corollaries. The paper also analyzes the necessity of hypotheses, discusses alternatives for computational efficiency, and provides bounds on missing points, contributing a foundational step toward surjective radical parametrizations and informing applications where full coverage is essential.

Abstract

In this paper, we introduce the notion of surjective radical parametrization and we prove sufficient conditions for a radical curve parametrization to be surjective.

Sufficient conditions for the surjectivity of radical curve parametrizations

TL;DR

This paper introduces a notion of surjectivity for radical curve parametrizations and provides sufficient conditions to guarantee that a radical parametrization covers the entire radical curve. By extending techniques from rational parametrizations, it uses a lifting strategy based on the Extension Theorem to propagate a point from the projection of the radical data to a full preimage, and introduces the notions of guilty and suspicious polynomials to manage degree losses caused by radical expressions. The main result identifies a concrete, checkable criterion: if there exists an index with not guilty and , and for all the ideals equal , then the parametrization is surjective; variations yield practical, computable corollaries. The paper also analyzes the necessity of hypotheses, discusses alternatives for computational efficiency, and provides bounds on missing points, contributing a foundational step toward surjective radical parametrizations and informing applications where full coverage is essential.

Abstract

In this paper, we introduce the notion of surjective radical parametrization and we prove sufficient conditions for a radical curve parametrization to be surjective.
Paper Structure (5 sections, 11 theorems, 34 equations)

This paper contains 5 sections, 11 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Sufficient condition for surjectivity
  4. On the hypotheses for surjectivity
  5. On missing points

Key Result

Theorem 2.5

[Extension Theorem, see CoxLittleOshea2015a Ch. 3.1, Th. 3] Let $I=\langle f_1,\ldots,f_s\rangle\subseteq\mathbb{C}[x_1,\ldots,x_n]$ and let $\widetilde{I}=I\cap\mathbb{C}[x_2,\ldots,x_n]$. For each $1\leq i\leq s$, write where $N_i\geq0$ and $c_i\neq0$. Suppose that we have a partial solution $(a_2,\ldots,a_n)\in V(\widetilde{I})$. If there exists $c_i$ which is nonzero at $(a_2,\ldots,a_n)$ the

Theorems & Definitions (49)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Example 2.4
  • Theorem 2.5
  • Definition 2.6
  • Example 2.7
  • Definition 2.8
  • Example 2.9: continuation of Example \ref{['ej:eje']}
  • Definition 3.1
  • ...and 39 more