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Register a new userInverting catalecticants of ternary quartics
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We study the reciprocal variety to the linear space of symmetric matrices (LSSM) of catalecticant matrices associated with ternary quartics. With numerical tools, we obtain 85 to be its degree and 36 to be the ML-degree of the LSSM. We provide a geometric explanation to why equality between these two invariants is not reached, as opposed to the case of binary forms, by describing the intersection of the reciprocal variety and the orthogonal of the LSSM in the rank loci. Moreover, we prove that only the rank-$1$ locus, namely the Veronese surface $ u_4(mathbb{P}^2)$, contributes to the degree of the reciprocal variety.
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Let F denote a homogeneous degree 4 polynomial in 3 variables, and let s be an integer between 1 and 5. We would like to know if F can be written as a sum of fourth powers of s linear forms (or a degeneration). We determine necessary and sufficient conditions for this to be possible. These conditions are expressed as the vanishing of certain concomitants of F for the natural action of SL_3.
Ternary real-valued quartics in $mathbb{R}^3$ being invariant under octahedral symmetry are considered. The geometric classification of these surfaces is given. A new type of surfaces emerge from this classification.
We show how one can use the representation theory of ternary quartics to construct all vector-valued Siegel modular forms and Teichmuller modular forms of degree 3. The relation between the order of vanishing of a concomitant on the locus of double conics and the order of vanishing of the corresponding modular form on the hyperelliptic locus plays an important role. We also determine the connection between Teichmuller cusp forms on overline{M}_g and the middle cohomology of symplectic local systems on M_g. In genus 3, we make this explicit in a large number of cases.
We propose faster methods for unconstrained optimization of emph{structured convex quartics}, which are convex functions of the form begin{equation*} f(x) = c^top x + x^top mathbf{G} x + mathbf{T}[x,x,x] + frac{1}{24} mathopen| mathbf{A} x mathclose|_4^4 end{equation*} for $c in mathbb{R}^d$, $mathbf{G} in mathbb{R}^{d times d}$, $mathbf{T} in mathbb{R}^{d times d times d}$, and $mathbf{A} in mathbb{R}^{n times d}$ such that $mathbf{A}^top mathbf{A} succ 0$. In particular, we show how to achieve an $epsilon$-optimal minimizer for such functions with only $O(n^{1/5}log^{O(1)}(mathcal{Z}/epsilon))$ calls to a gradient oracle and linear system solver, where $mathcal{Z}$ is a problem-dependent parameter. Our work extends recent ideas on efficient tensor methods and higher-order acceleration techniques to develop a descent method for optimizing the relevant quartic functions. As a natural consequence of our method, we achieve an overall cost of $O(n^{1/5}log^{O(1)}(mathcal{Z} / epsilon))$ calls to a gradient oracle and (sparse) linear system solver for the problem of $ell_4$-regression when $mathbf{A}^top mathbf{A} succ 0$, providing additional insight into what may be achieved for general $ell_p$-regression. Our results show the benefit of combining efficient higher-order methods with recent acceleration techniques for improving convergence rates in fundamental convex optimization problems.
Classical questions in extremal graph theory concern the asymptotics of $operatorname{ex}(G, mathcal{H})$ where $mathcal{H}$ is a fixed family of graphs and $G=G_n$ is taken from a `standard increasing sequence of host graphs $(G_1, G_2, dots)$, most often $K_n$ or $K_{n,n}$. Inverting the question, we can instead ask how large $e(G)$ can be with respect to $operatorname{ex}(G,mathcal{H})$. We show that the standard sequences indeed maximize $e(G)$ for some choices of $mathcal{H}$, but not for others. Many interesting questions and previous results arise very naturally in this context, which also, unusually, gives rise to sensible extremal questions concerning multigraphs and non-uniform hypergraphs.
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