The Algebraic Boundary of Graph Elliptopes
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914527773196288 |
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| author | Laurent, Monique Mascarin, Francesco Maria Telen, Simon |
| author_facet | Laurent, Monique Mascarin, Francesco Maria Telen, Simon |
| contents | This paper studies the algebraic boundary of the elliptope $\mathcal{E}(G)$ of a graph $G$. In particular, we completely characterize the algebraic boundary of $\mathcal{E}(G)$ when $G$ is cycle completable. In this case, the boundary is a union of determinantal hypersurfaces and Lissajous varieties, i.e., images of rational linear subspaces under the coordinatewise cosine map. As an application, we show that the algebraic boundary of $\mathcal{E}(G)$ is disjoint from its interior precisely when $\mathcal{E}(G)$ is a spectrahedron or, equivalently, when $G$ is a chordal graph. A central ingredient for the defining equation of the boundary hypersurface is the cycle polynomial, which captures the algebraic boundary of the elliptope $\mathcal{E}(C_n)$ of the $n$-th cycle graph $C_n$. We show that the cycle polynomial of $C_n$ is the resultant of two smaller cycle polynomials. Via this result, Sylvester's determinantal formula offers an inductive method for computing the cycle polynomial which mirrors a geometric property of metric polytopes. We also determine the degree of the homogeneous cycle polynomial, settling an open question of Sturmfels and Uhler (2010). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_02484 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Algebraic Boundary of Graph Elliptopes Laurent, Monique Mascarin, Francesco Maria Telen, Simon Algebraic Geometry This paper studies the algebraic boundary of the elliptope $\mathcal{E}(G)$ of a graph $G$. In particular, we completely characterize the algebraic boundary of $\mathcal{E}(G)$ when $G$ is cycle completable. In this case, the boundary is a union of determinantal hypersurfaces and Lissajous varieties, i.e., images of rational linear subspaces under the coordinatewise cosine map. As an application, we show that the algebraic boundary of $\mathcal{E}(G)$ is disjoint from its interior precisely when $\mathcal{E}(G)$ is a spectrahedron or, equivalently, when $G$ is a chordal graph. A central ingredient for the defining equation of the boundary hypersurface is the cycle polynomial, which captures the algebraic boundary of the elliptope $\mathcal{E}(C_n)$ of the $n$-th cycle graph $C_n$. We show that the cycle polynomial of $C_n$ is the resultant of two smaller cycle polynomials. Via this result, Sylvester's determinantal formula offers an inductive method for computing the cycle polynomial which mirrors a geometric property of metric polytopes. We also determine the degree of the homogeneous cycle polynomial, settling an open question of Sturmfels and Uhler (2010). |
| title | The Algebraic Boundary of Graph Elliptopes |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2605.02484 |