arXiv subjects

Thi Xuan Vu

Publications and source records attributed to Thi Xuan Vu.

13 recordsLinked to original sources

arXiv2026

A Symbolic Homotopy Algorithm for Solving Composable Polynomial Systems

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 $E_6, E_7, E_8, F_4, H_3, H_4$.

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arXiv2025

Symbolic Computation with Symmetric Polynomials in Real Algebraic Geometry

Symmetry plays a central role in accelerating symbolic computation involving polynomials. This chapter surveys recent developments and foundational methods that leverage the inherent symmetries of polynomial systems to reduce complexity, improve algorithmic efficiency, and reveal deeper structural insights. The main focus is on symmetry by the permutation of variables.

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arXiv2025

Computing Polynomial Representation in Subrings of Multivariate Polynomial Rings

Let $\mathcal{R} = \mathbb{K}[x_1, \dots, x_n]$ be a multivariate polynomial ring over a field $\mathbb{K}$ of characteristic 0. Consider $n$ algebraically independent elements $g_1, \dots, g_n$ in $\mathcal{R}$. Let $\mathcal{S}$ denote the subring of $\mathcal{R}$ generated by $g_1, \dots, g_n$, and let $h$ be an element of $\mathcal{S}$. Then, there exists a unique element ${f} \in \mathbb{K}[u_1, \dots, u_n]$ such that $h = f(g_1, \dots, g_n)$. In this paper, we provide an algorithm for computing ${f}$, given $h$ and $g_1, \dots, g_n$. The complexity of our algorithm is linear in the size of the input, $h$ and $g_1, \dots, g_n$, and polynomial in $n$ when the degree of $f$ is fixed. Previous works are mostly known when $f$ is a symmetric polynomial and $g_1, \dots, g_n$ are elementary symmetric, homogeneous symmetric, or power symmetric polynomials.

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arXiv2025

Deciding Connectivity in Symmetric Semi-Algebraic Sets

A semi-algebraic set is a subset of $\mathbb{R}^n$ defined by a finite collection of polynomial equations and inequalities. In this paper, we investigate the problem of determining whether two points in such a set belong to the same connected component. We focus on the case where the defining equations and inequalities are invariant under the natural action of the symmetric group and where each polynomial has degree at most $ d $, with $ d < n $ (where $ n $ denotes the number of variables). Exploiting this symmetry, we develop and analyze algorithms for two key tasks. First, we present an algorithm that determines whether the orbits of two given points are connected. Second, we provide an algorithm that decides connectivity between arbitrary points in the set. Both algorithms run in polynomial time with respect to $ n $.

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arXiv2024

Connectivity in Symmetric Semi-Algebraic Sets

Semi-algebraic set is a subset of the real space defined by polynomial equations and inequalities. In this paper, we consider the problem of deciding whether two given points in a semi-algebraic set are connected. We restrict to the case when all equations and inequalities are invariant under the action of the symmetric group and their degrees at most $d<n$, where $n$ is the number of variables. Additionally, we assume that the two points are in the same fundamental domain of the action of the symmetric group, by assuming that the coordinates of two given points are sorted in non-decreasing order. We construct and analyze an algorithm that solves this problem, by taking advantage of the group action, and has a complexity being polynomial in $n$.

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arXiv2023

Faster real root decision algorithm for symmetric polynomials

In this paper, we consider the problem of deciding the existence of real solutions to a system of polynomial equations having real coefficients, and which are invariant under the action of the symmetric group. We construct and analyze a Monte Carlo probabilistic algorithm which solves this problem, under some regularity assumptions on the input, by taking advantage of the symmetry invariance property. The complexity of our algorithm is polynomial in $d^s, {{n+d} \choose d}$, and ${{n} \choose {s+1}}$, where $n$ is the number of variables and $d$ is the maximal degree of $s$ input polynomials defining the real algebraic set under study. In particular, this complexity is polynomial in $n$ when $d$ and $s$ are fixed and is equal to $n^{O(1)}2^n$ when $d=n$.

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arXiv2022

Computing critical points for algebraic systems defined by hyperoctahedral invariant polynomials

Let $\mathbb{K}$ be a field of characteristic zero and $\mathbb{K}[x_1, \dots, x_n]$ the corresponding multivariate polynomial ring. Given a sequence of $s$ polynomials $\mathbf{f} = (f_1, \dots, f_s)$ and a polynomial $ϕ$, all in $\mathbb{K}[x_1, \dots, x_n]$ with $s<n$, we consider the problem of computing the set $W(ϕ, \mathbf{f})$ of points at which $\mathbf{f}$ vanishes and the Jacobian matrix of $\mathbf{f}, ϕ$ with respect to $x_1, \dots, x_n$ does not have full rank. This problem plays an essential role in many application areas. In this paper we focus on a case where the polynomials are all invariant under the action of the signed symmetric group $B_n$. We introduce a notion called {\em hyperoctahedral representation} to describe $B_n$-invariant sets. We study the invariance properties of the input polynomials to split $W(ϕ, \mathbf{f})$ according to the orbits of $B_n$ and then design an algorithm whose output is a {hyperoctahedral representation} of $W(ϕ, \mathbf{f})$. The runtime of our algorithm is polynomial in the total number of points described by the output.

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arXiv2022

On the complexity of invariant polynomials under the action of finite reflection groups

Let $\mathbb{K}[x_1, \dots, x_n]$ be a multivariate polynomial ring over a field $\mathbb{K}$. Let $(u_1, \dots, u_n)$ be a sequence of $n$ algebraically independent elements in $\mathbb{K}[x_1, \dots, x_n]$. Given a polynomial $f$ in $\mathbb{K}[u_1, \dots, u_n]$, a subring of $\mathbb{K}[x_1, \dots, x_n]$ generated by the $u_i$'s, we are interested infinding the unique polynomial $f_{\rm new}$ in $\mathbb{K}[e_1,\dots, e_n]$, where $e_1, \dots, e_n$ are new variables, such that $f_{\mathrm{new}}(u_1, \dots, u_n) = f(x_1, \dots, x_n)$. We provide an algorithm and analyze its arithmetic complexity to compute $f_{\mathrm{new}}$ knowing $f$ and $(u_1, \dots, u_n)$.

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arXiv2022

Rank-Sensitive Computation of the Rank Profile of a Polynomial Matrix

Consider a matrix $\mathbf{F} \in \mathbb{K}[x]^{m \times n}$ of univariate polynomials over a field $\mathbb{K}$. We study the problem of computing the column rank profile of $\mathbf{F}$. To this end we first give an algorithm which improves the minimal kernel basis algorithm of Zhou, Labahn, and Storjohann (Proceedings ISSAC 2012). We then provide a second algorithm which computes the column rank profile of $\mathbf{F}$ with a rank-sensitive complexity of $O\tilde{~}(r^{ω-2} n (m+D))$ operations in $\mathbb{K}$. Here, $D$ is the sum of row degrees of $\mathbf{F}$, $ω$ is the exponent of matrix multiplication, and $O\tilde{~}(\cdot)$ hides logarithmic factors.

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arXiv2020

Homotopy techniques for solving sparse column support determinantal polynomial systems

Let $\mathbf{K}$ be a field of characteristic zero with $\overline{\mathbf{K}}$ its algebraic closure. Given a sequence of polynomials $\mathbf{g} = (g_1, \ldots, g_s) \in \mathbf{K}[x_1, \ldots , x_n]^s$ and a polynomial matrix $\mathbf{F} = [f_{i,j}] \in \mathbf{K}[x_1, \ldots, x_n]^{p \times q}$, with $p \leq q$, we are interested in determining the isolated points of $V_p(\mathbf{F},\mathbf{g})$, the algebraic set of points in $\overline{\mathbf{K}}$ at which all polynomials in $\mathbf{g}$ and all $p$-minors of $\mathbf{F}$ vanish, under the assumption $n = q - p + s + 1$. Such polynomial systems arise in a variety of applications including for example polynomial optimization and computational geometry. We design a randomized sparse homotopy algorithm for computing the isolated points in $V_p(\mathbf{F},\mathbf{g})$ which takes advantage of the determinantal structure of the system defining $V_p(\mathbf{F}, \mathbf{g})$. Its complexity is polynomial in the maximum number of isolated solutions to such systems sharing the same sparsity pattern and in some combinatorial quantities attached to the structure of such systems. It is the first algorithm which takes advantage both on the determinantal structure and sparsity of input polynomials. We also derive complexity bounds for the particular but important case where $\mathbf{g}$ and the columns of $\mathbf{F}$ satisfy weighted degree constraints. Such systems arise naturally in the computation of critical points of maps restricted to algebraic sets when both are invariant by the action of the symmetric group.

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arXiv2020

Computing critical points for invariant algebraic systems

Let $\mathbf{K}$ be a field and $ϕ$, $\mathbf{f} = (f_1, \ldots, f_s)$ in $\mathbf{K}[x_1, \dots, x_n]$ be multivariate polynomials (with $s < n$) invariant under the action of $\mathcal{S}_n$, the group of permutations of $\{1, \dots, n\}$. We consider the problem of computing the points at which $\mathbf{f}$ vanish and the Jacobian matrix associated to $\mathbf{f}, ϕ$ is rank deficient provided that this set is finite. We exploit the invariance properties of the input to split the solution space according to the orbits of $\mathcal{S}_n$. This allows us to design an algorithm which gives a triangular description of the solution space and which runs in time polynomial in $d^s$, ${{n+d}\choose{d}}$ and $\binom{n}{s+1}$ where $d$ is the maximum degree of the input polynomials. When $d,s$ are fixed, this is polynomial in $n$ while when $s$ is fixed and $d \simeq n$ this yields an exponential speed-up with respect to the usual polynomial system solving algorithms.

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arXiv2018

Solving determinantal systems using homotopy techniques

Let $\K$ be a field of characteristic zero and $\Kbar$ be an algebraic closure of $\K$. Consider a sequence of polynomials$G=(g\_1,\dots,g\_s)$ in $\K[X\_1,\dots,X\_n]$, a polynomial matrix $\F=[f\_{i,j}] \in \K[X\_1,\dots,X\_n]^{p \times q}$, with $p \leq q$,and the algebraic set $V\_p(F, G)$ of points in $\KKbar$ at which all polynomials in $\G$ and all $p$-minors of $\F$vanish. Such polynomial systems appear naturally in e.g. polynomial optimization, computational geometry.We provide bounds on the number of isolated points in $V\_p(F, G)$ depending on the maxima of the degrees in rows (resp. columns) of $\F$. Next, we design homotopy algorithms for computing those points. These algorithms take advantage of the determinantal structure of the system defining $V\_p(F, G)$. In particular, the algorithms run in time that is polynomial in the bound on the number of isolated points.

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arXiv2017

Computing Canonical Bases of Modules of Univariate Relations

We study the computation of canonical bases of sets of univariate relations $(p_1,\ldots,p_m) \in \mathbb{K}[x]^{m}$ such that $p_1 f_1 + \cdots + p_m f_m = 0$; here, the input elements $f_1,\ldots,f_m$ are from a quotient $\mathbb{K}[x]^n/\mathcal{M}$, where $\mathcal{M}$ is a $\mathbb{K}[x]$-module of rank $n$ given by a basis $\mathbf{M}\in\mathbb{K}[x]^{n\times n}$ in Hermite form. We exploit the triangular shape of $\mathbf{M}$ to generalize a divide-and-conquer approach which originates from fast minimal approximant basis algorithms. Besides recent techniques for this approach, we rely on high-order lifting to perform fast modular products of polynomial matrices of the form $\mathbf{P}\mathbf{F} \bmod \mathbf{M}$. Our algorithm uses $O\tilde{~}(m^{ω-1}D + n^ω D/m)$ operations in $\mathbb{K}$, where $D = \mathrm{deg}(\det(\mathbf{M}))$ is the $\mathbb{K}$-vector space dimension of $\mathbb{K}[x]^n/\mathcal{M}$, $O\tilde{~}(\cdot)$ indicates that logarithmic factors are omitted, and $ω$ is the exponent of matrix multiplication. This had previously only been achieved for a diagonal matrix $\mathbf{M}$. Furthermore, our algorithm can be used to compute the shifted Popov form of a nonsingular matrix within the same cost bound, up to logarithmic factors, as the previously fastest known algorithm, which is randomized.

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