Ideals of equations for elements in a free group and context-free languages
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913280181665792 |
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| author | Ascari, Dario |
| author_facet | Ascari, Dario |
| contents | Let $F$ be a finitely generated free group, and let $H\le F$ be a finitely generated subgroup. An equation for an element $g\in F$ with coefficients in $H$ is an element $w(x)\in H*\langle x \rangle$ such that $w(g)=1$ in $F$; the degree of the equation is the number of occurrences of $x$ and $x^{-1}$ in the cyclic reduction of $w(x)$. Given an element $g\in F$, we consider the ideal $\mathfrak{I}_g\subseteq H*\langle x \rangle$ of equations for $g$ with coefficients in $H$; we study the structure of $\mathfrak{I}_g$ using context-free languages.
We describe a new algorithm that determines whether $\mathfrak{I}_g$ is trivial or not; the algorithm runs in polynomial time. We also describe a polynomial-time algorithm that, given $d\in\mathbb{N}$, decides whether or not the subset $\mathfrak{I}_{g,d}\subseteq\mathfrak{I}_g$ of all degree-$d$ equations is empty. We provide a polynomial-time algorithm that computes the minimum degree $d_{\min}$ of a non-trivial equation in $\mathfrak{I}_g$. We provide a sharp upper bound on $d_{\min}$. Finally, we study the growth of the number of (cyclically reduced) equations in $\mathfrak{I}_g$ and in $\mathfrak{I}_{g,d}$ as a function of their length. We prove that this growth is either polynomial or exponential, and we provide a polynomial-time algorithm that computes the type of growth (including the degree of the growth if it's polynomial). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_10276 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Ideals of equations for elements in a free group and context-free languages Ascari, Dario Group Theory 20F70, 20E05 (Primary) 20F10, 20E07 (Secondary) Let $F$ be a finitely generated free group, and let $H\le F$ be a finitely generated subgroup. An equation for an element $g\in F$ with coefficients in $H$ is an element $w(x)\in H*\langle x \rangle$ such that $w(g)=1$ in $F$; the degree of the equation is the number of occurrences of $x$ and $x^{-1}$ in the cyclic reduction of $w(x)$. Given an element $g\in F$, we consider the ideal $\mathfrak{I}_g\subseteq H*\langle x \rangle$ of equations for $g$ with coefficients in $H$; we study the structure of $\mathfrak{I}_g$ using context-free languages. We describe a new algorithm that determines whether $\mathfrak{I}_g$ is trivial or not; the algorithm runs in polynomial time. We also describe a polynomial-time algorithm that, given $d\in\mathbb{N}$, decides whether or not the subset $\mathfrak{I}_{g,d}\subseteq\mathfrak{I}_g$ of all degree-$d$ equations is empty. We provide a polynomial-time algorithm that computes the minimum degree $d_{\min}$ of a non-trivial equation in $\mathfrak{I}_g$. We provide a sharp upper bound on $d_{\min}$. Finally, we study the growth of the number of (cyclically reduced) equations in $\mathfrak{I}_g$ and in $\mathfrak{I}_{g,d}$ as a function of their length. We prove that this growth is either polynomial or exponential, and we provide a polynomial-time algorithm that computes the type of growth (including the degree of the growth if it's polynomial). |
| title | Ideals of equations for elements in a free group and context-free languages |
| topic | Group Theory 20F70, 20E05 (Primary) 20F10, 20E07 (Secondary) |
| url | https://arxiv.org/abs/2211.10276 |