The second moment of the number of integral points on elliptic curves is bounded

In this paper, we show that the second moment of the number of integral points on elliptic curves over $\mathbb{Q}$ is bounded. In particular, we prove that, for any $0 < s < \log_2 5 = 2.3219 \ldots$, the $s$-th moment of the number of integral points is bounded for many families of elliptic curves --- e.g., for the family of all integral short Weierstrass curves ordered by naive height, for the family of only minimal such Weierstrass curves, for the family of semistable curves, or for subfamilies thereof defined by finitely many congruence conditions. For certain other families of elliptic curves, such as those with a marked point or a marked $2$-torsion point, the same methods show that for $0 < s < \log_2 3 = 1.5850\ldots$, the $s$-th moment of the number of integral points is bounded. The main new ingredient in our proof is an upper bound on the number of integral points on an affine integral Weierstrass model of an elliptic curve depending only on the rank of the curve and the number of square divisors of the discriminant. We obtain the bound by studying a bijection first observed by Mordell between integral points on these curves and certain types of binary quartic forms. The theorems on moments then follow from H\"older's inequality, analytic techniques, and results on bounds on the average sizes of Selmer groups in the families.Comment: 14 pages, comments welcome

Reconstructing general plane quartics from their inflection lines

Let $C$ be a general plane quartic and let ${\rm Fl}(C)$ denote the configuration of inflection lines of $C$. We show that if $D$ is any plane quartic with the same configuration of inflection lines ${\rm Fl}(C)$, then the quartics $C$ and $D$ coincide.Comment: 21 pages, to appear in Transactions of the American Mathematical Societ

Apolarity, Hessian and Macaulay polynomials

A result by Macaulay states that an Artinian graded Gorenstein ring R of socle dimension one and socle degree b can be realized as the apolar ring of a homogeneous polynomial f of degree b. If R is the Jacobian ring of a smooth hypersurface g=0, then b is just equal to the degree of the Hessian polynomial of g. In this paper we investigate the relationship between f and the Hessian polynomial of g.Comment: 12 pages. Improved exposition, minor correction

On nonsingular two-step nilpotent Lie algebras

A 2-step nilpotent Lie algebra n is called nonsingular if ad(X): n --> [n,n] is onto for any X not in [n,n]. We explore nonsingular algebras in several directions, including the classification problem (isomorphism invariants), the existence of canonical inner products (nilsolitons) and their automorphism groups (maximality properties). Our main tools are the moment map for certain real reductive representations, and the Pfaffian form of a 2-step algebra, which is a positive homogeneous polynomial in the nonsingular case.Comment: 26 pages. Final version to appear in Math Res Let

Computing Linear Matrix Representations of Helton-Vinnikov Curves

Helton and Vinnikov showed that every rigidly convex curve in the real plane bounds a spectrahedron. This leads to the computational problem of explicitly producing a symmetric (positive definite) linear determinantal representation for a given curve. We study three approaches to this problem: an algebraic approach via solving polynomial equations, a geometric approach via contact curves, and an analytic approach via theta functions. These are explained, compared, and tested experimentally for low degree instances.Comment: 19 pages, 3 figures, minor revisions; Mathematical Methods in Systems, Optimization and Control, Birkhauser, Base

An elementary proof of Hilbert's theorem on ternary quartics

In 1888, Hilbert proved that every non-negative quartic form f=f(x,y,z) with real coefficients is a sum of three squares of quadratic forms. His proof was ahead of its time and used advanced methods from topology and algebraic geometry. Up to now, no elementary proof is known. Here we present a completely new approach. Although our proof is not easy, it uses only elementary techniques. As a by-product, it gives information on the number of representations f=p_1^2+p_2^2+p_3^2 of f up to orthogonal equivalence. We show that this number is 8 for generically chosen f, and that it is 4 when f is chosen generically with a real zero. Although these facts were known, there was no elementary approach to them so far.Comment: 26 page