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Register a new userA curious identity that implies Fabers conjecture
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We prove that a curious generating series identity implies Fabers intersection number conjecture (by showing that it implies a combinatorial identity already given in arXiv:1902.02742) and give a new proof of Fabers conjecture by directly proving this identity.
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We employ the $1/2$-spin tautological relations to provide a particular combinatorial identity. We show that this identity is a statement equivalent to Fabers formula for proportionalities of kappa-classes on $mathcal{M}_g$, $ggeq 2$. We then prove several cases of the combinatorial identity, providing a new proof of Fabers formula for those cases.
We give a criterion for modular extension of rank-4 hypermodular matroids, and prove a weakening of Kantors conjecture for rank-4 realizable matroids. This proves the sticky matroid conjecture and Kantors conjecture for realizable matroids due to an argument of Bachem, Kern, and Bonin, and due to an equivalence argument of Hochstattler and Wilhelmi, respectively.
We show Kantors conjecture (1974) holds in rank 4. This proves both the sticky matroid conjecture of Poljak and Turzik (1982) and the whole Kantors conjecture, due to an argument of Bachem, Kern, and Bonin, and an equivalence argument of Hochstattler and Wilhelmi, respectively.
A well-known combinatorial theorem says that a set of n non-collinear points in the plane determines at least n distinct lines. Chen and Chvatal conjectured that this theorem extends to metric spaces, with an appropriated definition of line. In this work we prove a slightly stronger version of Chen and Chvatal conjecture for a family of graphs containing chordal graphs and distance-hereditary graphs.
Let $p_n(x)$, $n=0,1,dots$, be the orthogonal polynomials with respect to a given density $dmu(x)$. Furthermore, let $d u(x)$ be a density which arises from $dmu(x)$ by multiplication by a rational function in $x$. We prove a formula that expresses the Hankel determinants of moments of $d u(x)$ in terms of a determinant involving the orthogonal polynomials $p_n(x)$ and associated functions $q_n(x)=int p_n(u) ,dmu(u)/(x-u)$. Uvarovs formula for the orthogonal polynomials with respect to $d u(x)$ is a corollary of our theorem. Our result generalises a Hankel determinant formula for the case where the rational function is a polynomial that existed somehow hidden in the folklore of the theory of orthogonal polynomials but has been stated explicitly only relatively recently (see [arXiv:2101.04225]). Our theorem can be interpreted in a two-fold way: analytically or in the sense of formal series. We apply our theorem to derive several curious Hankel determinant evaluations.
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