Corrigendum for "Almost vanishing polynomials and an application to the Hough transform"

TL;DR

This paper corrects a technical error in a previous work on almost vanishing polynomials and their application to the Hough transform, updating the bounds while preserving the original proofs' structure.

Contribution

It provides a correction to the bounds in the original paper, ensuring the accuracy of the theoretical results related to almost vanishing polynomials.

Findings

Corrected bounds in the original theorem

Maintained the original proof structure

Ensured the validity of the application to the Hough transform

Abstract

In this note we correct a technical error occurred in [M. Torrente and M.C. Beltrametti, "Almost vanishing polynomials and an application to the Hough transform", J. Algebra Appl. 13(8), (2014)]. This affects the bounds given in that paper, even though the structure and the logic of all proofs remain fully unchanged.

Figures

Equations

$$|{\mathrm{Jac}}_f(p)(p^{\ast }-p{)}^t|$$

$$|f(p)|>\Vert {\mathrm{Jac}}_f(p{)}^t{\Vert }_1{\mathbf{\epsilon }}_{\max }+\frac{n}{2}\mathsf{H}{\mathbf{\epsilon }}_{\max }^2=:B_1,$$

$$|f(p)|>\Vert {\mathrm{Jac}}_f(p{)}^t{\Vert }_1{\mathbf{\epsilon }}_{\max }+\frac{n}{2}\Vert H_f(p){\Vert }_{\infty }{\mathbf{\epsilon }}_{\max }^2:=B_1^{\prime },$$

$$|f(p)|<\frac{2R}{\mathsf{J}(c+n^{5/2}\mathsf{H}\mathsf{J}R)}=:B_2,$$

$$\Theta :=\Vert {\mathrm{Jac}}_f^{†}(p){\Vert }_{\infty }+n^2(1+2\sqrt{n})\frac{\Vert H_f(p){\Vert }_{\infty }}{\Vert {\mathrm{Jac}}_f(p){\Vert }_1^2}R.$$

$$|f(p)|<\frac{2R}{\Theta (c+n^{9/2}\Vert H_f(p){\Vert }_{\infty }\Theta R)}=:B_2^{\prime },$$

Full text

Corrigendum for “Almost vanishing polynomials

and an application to the Hough transform” 1112010 Mathematics Subject Classification. Primary 26C10, 15A60, 14Q10; Secondary 14H50

Keywords and phrases. Weighted matrix norm; hypersurface; crossing area conditions

Maria-Laura Torrente and Mauro C. Beltrametti

In this note we correct a technical error occurred in [1]. This affects the bounds given in that paper, even though the structure and the logic of all proofs remain fully unchanged. The error is due to a repeated wrong use of Hölder’s inequality (a transpose of a matrix was missed). The first time it occurs is in the first inequality of formula $(8)$ of [1]. Indeed, the correct version of that formula is:

[TABLE]

We refer to our paper [1], using the same notation. Here, we confine ourselves to state the correct versions of Proposition 2.1, Proposition 2.5, Proposition 3.2, and Proposition 4.3 of [1], respectively. Up to the error pointed out and corrected as above, the proofs go parallel to those in [1]. In the following, $P$ denotes the multivariate polynomial ring ${\mathbb{R}}[{\bm{x}}]={\mathbb{R}}[x_{1},\ldots,x_{n}]$.

Proposition 2.1 Let $f=f({\bm{x}})$ be a non-constant polynomial of $P$, let $p$ be a point of ${\mathbb{R}}^{n}$, and let ${\bf C}(p)\subseteq{\bf B}(p)$ be an $(\infty,{\bm{\varepsilon}})$-unit cell centered at $p$. If

[TABLE]

*then the hypersurface of equation $f=0$ does not cross ${\bf C}(p).$ *

Proposition 2.5 Let $f({\bm{x}})$ be a degree $\geq 2$ polynomial of $P$. Let $p$ be a point of ${\mathbb{R}}^{n}$ and let ${\bf C}(p)\subseteq{\bf B}(p)$ be an $(\infty,{\bm{\varepsilon}})$-unit cell centered at $p$. If

[TABLE]

*then the hypersurface of equation $f=0$ does not cross ${\bf C}(p)$ neglecting contributions of order ${\rm O}({\bm{\varepsilon}}_{\rm max}^{3})$. *

Proposition 3.2 Let $f=f({\bm{x}})$ be a degree $\geq 2$ polynomial of $P$, let $p$ be a point of ${\mathbb{R}}^{n}$ such that $\mathop{\rm Jac}\nolimits_{f}(p)$ is not the zero vector, and let ${\bf C}(p)\subseteq{\bf B}(p)$ be an $(\infty,{\bm{\varepsilon}})$-unit cell centered at $p$. Let $R$ be a positive real number such that R<\min\big{\{}{\bm{\varepsilon}}_{\rm min},\frac{\|\mathop{\rm Jac}\nolimits_{f}(p)\|_{1}}{{\mathsf{H}}}\big{\}}. Set $c:=\max\{2,\sqrt{n}\}$. If

[TABLE]

*then the hypersurface of equation $f=0$ crosses ${\bf C}(p)$. *

Proposition 4.3 Let $f=f({\bm{x}})$ be a degree $\geq 2$ polynomial of $P$, let $p$ be a point of ${\mathbb{R}}^{n}$ such that Jacobian $\mathop{\rm Jac}\nolimits_{f}(p)$ and the Hessian matrix $H_{f}(p)$ are nontrivial, and let ${\bf C}(p)\subseteq{\bf B}(p)$ be an $(\infty,{\bm{\varepsilon}})$-unit cell centered at $p$. Let $R$ be a positive real number such that $R<\min\left\{{\bm{\varepsilon}}_{\rm min},\frac{\|\mathop{\rm Jac}\nolimits_{f}(p)\|_{1}}{n^{2}\|H_{f}(p)\|_{\infty}}\right\}$, let $c:=\max\{2,\sqrt{n}\}$ and set

[TABLE]

If

[TABLE]

*then the hypersurface of equation $f=0$ crosses ${\bf C}(p)$ neglecting order ${\rm O}(R^{2})$ contributions. *

Remark. More accurate and general estimates [2], when specialized to the case of hypersurfaces and $\infty$-norm, allow us to improve the bounds above, this also assuring that the applications discussed in [1, Section 6] still remain meaningful. Precisely, in [2, Theorem 3.2], we can in fact show that the bound $B_{2}$ goes as $\frac{1}{n}$ instead of $\frac{1}{n^{5/2}}$, weakening the assumption. Similarly, one can shows that the bound $B_{2}^{\prime}$ in [2, Theorem 4.6] goes as $\frac{1}{n^{5/2}}$ instead of $\frac{1}{n^{9/2}}$.