Certificate for Orthogonal Equivalence of Real Polynomials by Polynomial-Weighted Principal Component Analysis
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866918281068150784 |
|---|---|
| author | Helmer, Martin Hong, David Hong, Hoon |
| author_facet | Helmer, Martin Hong, David Hong, Hoon |
| contents | Suppose that $f(x) \in \mathbb{R}[x_1,\dots, x_n]$ and $g(x) \in \mathbb{R}[x_1,\dots, x_n]$ are two real polynomials of degree $d$ in $n$ variables. If the polynomials $f$ and $g$ are the same up to orthogonal symmetry a natural question is then what element of the orthogonal group induces the orthogonal symmetry; i.e. to find the element $R\in O(n)$ such that $f(Rx)=g(x)$. One may directly solve this problem by constructing a nonlinear system of equations induced by the relation $f(Rx)=g(x)$ along with the identities of the orthogonal group however this approach becomes quite computationally expensive for larger values of $n$ and $d$. To give an alternative and significantly more scalable solution to this problem, we introduce the concept of Polynomial-Weighted Principal Component Analysis (PW-PCA). We in particular show how PW-PCA can be effectively computed and how these techniques can be used to obtain a certificate of orthogonal equivalence, that is we find the $R\in O(n)$ such that $f(Rx)=g(x)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_06148 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Certificate for Orthogonal Equivalence of Real Polynomials by Polynomial-Weighted Principal Component Analysis Helmer, Martin Hong, David Hong, Hoon Rings and Algebras Numerical Analysis Commutative Algebra Suppose that $f(x) \in \mathbb{R}[x_1,\dots, x_n]$ and $g(x) \in \mathbb{R}[x_1,\dots, x_n]$ are two real polynomials of degree $d$ in $n$ variables. If the polynomials $f$ and $g$ are the same up to orthogonal symmetry a natural question is then what element of the orthogonal group induces the orthogonal symmetry; i.e. to find the element $R\in O(n)$ such that $f(Rx)=g(x)$. One may directly solve this problem by constructing a nonlinear system of equations induced by the relation $f(Rx)=g(x)$ along with the identities of the orthogonal group however this approach becomes quite computationally expensive for larger values of $n$ and $d$. To give an alternative and significantly more scalable solution to this problem, we introduce the concept of Polynomial-Weighted Principal Component Analysis (PW-PCA). We in particular show how PW-PCA can be effectively computed and how these techniques can be used to obtain a certificate of orthogonal equivalence, that is we find the $R\in O(n)$ such that $f(Rx)=g(x)$. |
| title | Certificate for Orthogonal Equivalence of Real Polynomials by Polynomial-Weighted Principal Component Analysis |
| topic | Rings and Algebras Numerical Analysis Commutative Algebra |
| url | https://arxiv.org/abs/2601.06148 |