A Symbolic Homotopy Algorithm for Solving Composable Polynomial Systems
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866913152787021824 |
|---|---|
| author | Vu, Thi Xuan |
| author_facet | Vu, Thi Xuan |
| contents | We study the problem of computing the isolated regular solutions of a system \((f_1,\ldots,f_n)\) of \(n\) polynomial equations in \(n\) variables \((X_1, \dots, X_n)\) over a field of characteristic zero \(k\). We focus on systems with a \emph{composable structure}, where each polynomial \(f_i\) can be expressed as a composition \( f_i = h_i(g_1,\dots,g_n)\). Exploiting this structure allows us to reduce the original system to one in the \(g_j\) variables, thereby significantly improving the efficiency of symbolic solution algorithms. We present a probabilistic algorithm that computes all isolated regular solutions, with arithmetic complexity being polynomial in the input size and in the number of solutions.
A first important application is when \(f_1, \dots, f_n\) belong to the subring \(k[g_1, \dots, g_n]\), where \(g_1, \dots, g_n\) are algebraically independent polynomials in \(k[X_1, \dots, X_n]\). Another important application is to systems of invariant polynomials under finite reflection groups, since by the Chevalley-Shephard-Todd theorem their invariant rings are polynomial algebras. Typical examples include the symmetric groups \(S_n\), the hyperoctahedral groups \(B_n\), the dihedral groups \(I_2(m)\), and the exceptional finite reflection groups \(E_6, E_7, E_8, F_4, H_3, H_4\). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_22514 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Symbolic Homotopy Algorithm for Solving Composable Polynomial Systems Vu, Thi Xuan Symbolic Computation Commutative Algebra Algebraic Geometry We study the problem of computing the isolated regular solutions of a system \((f_1,\ldots,f_n)\) of \(n\) polynomial equations in \(n\) variables \((X_1, \dots, X_n)\) over a field of characteristic zero \(k\). We focus on systems with a \emph{composable structure}, where each polynomial \(f_i\) can be expressed as a composition \( f_i = h_i(g_1,\dots,g_n)\). Exploiting this structure allows us to reduce the original system to one in the \(g_j\) variables, thereby significantly improving the efficiency of symbolic solution algorithms. We present a probabilistic algorithm that computes all isolated regular solutions, with arithmetic complexity being polynomial in the input size and in the number of solutions. A first important application is when \(f_1, \dots, f_n\) belong to the subring \(k[g_1, \dots, g_n]\), where \(g_1, \dots, g_n\) are algebraically independent polynomials in \(k[X_1, \dots, X_n]\). Another important application is to systems of invariant polynomials under finite reflection groups, since by the Chevalley-Shephard-Todd theorem their invariant rings are polynomial algebras. Typical examples include the symmetric groups \(S_n\), the hyperoctahedral groups \(B_n\), the dihedral groups \(I_2(m)\), and the exceptional finite reflection groups \(E_6, E_7, E_8, F_4, H_3, H_4\). |
| title | A Symbolic Homotopy Algorithm for Solving Composable Polynomial Systems |
| topic | Symbolic Computation Commutative Algebra Algebraic Geometry |
| url | https://arxiv.org/abs/2605.22514 |