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We explore the concept of projections of syzygies and prove two new technical results; we firstly give a precise characterization of syzygy schemes in terms of their projections, secondly, we prove a converse to Aprodu's Projection Theorem.…

In this paper we generalize the study of Matiyasevich on integer points over conics, introducing the more general concept of radical points. With this generalization we are able to solve in positive integers some Diophantine equations,…

We consider polynomials on the intersection of the closed positive orthant with the height-$1$ level hypersurface of certain polynomials with positive coefficients. We show that any polynomial strictly positive on such a semi-algebraic set…

Let $(X,L)$ be a polarized K3 surface of genus $g$ and $C_{en} \subset X$ be the curve of singular points of nodal elliptic curves in $|L|$. When $(X,L)$ is generic of genus two, Huybrechts observed that the curve $C_{en}$ is a constant…

Let K be a convex body in $R^d$. A random polytope is the convex hull $[x_1,...,x_n]$ of finitely many points chosen at random in K. $\Bbb E(K,n)$ is the expectation of the volume of a random polytope of n randomly chosen points. I.…

Let $E$ be a non-CM elliptic curve defined over $\mathbb {Q}$. Fix an algebraic closure $\overline{\mathbb {Q}}$ of $\mathbb {Q}$. We get a Galois representation \[\rho_E \colon Gal(\overline{\mathbb {Q}}/\mathbb {Q}) \to GL_2(\hat{\mathbb…

Let P(x,y) be a rational polynomial and k in Q be a generic value. If the curve (P(x,y)=k) is irreducible and admits an infinite number of points whose coordinates are integers then there exist algebraic automorphisms that send P(x,y) to…

In this paper we obtain an explicit formula for the number of curves in a compact complex surface $X$ (passing through the right number of generic points), that has up to one node and one singularity of codimension $k$, provided the total…

We obtain local and global results on polynomially convex hulls of Lagrangian and totally real submanifolds of $C^n$ with self-intersections and open Whitney umbrella points.

We study some systems of polynomials whose support lies in the convex hull of a circuit, giving a sharp upper bound for their numbers of real solutions. This upper bound is non-trivial in that it is smaller than either the Kouchnirenko or…

If $E$ is an elliptic curve over $\mathbb{Q}$, then it follows from work of Serre and Hooley that, under the assumption of the Generalized Riemann Hypothesis, the density of primes $p$ such that the group of $\mathbb{F}_p$-rational points…

We study the relationship between singularities of finite-dimensional integrable systems and singularities of the corresponding spectral curves. For the large class of integrable systems on matrix polynomials, which is a general framework…

We study non-commutative degenerations of elliptic curves over local fields. The corresponding objects are close relatives of non-commutative tori of Connes and Rieffel.

We show that every split del Pezzo surface of degree d=5,4,3 or 2 has a universal torsor which is a dense open subset of the intersection of 6-d dilatations of the affine cone over the corresponding generalized Grassmannian G/P. Here a…

We describe the practical implementation of an average polynomial-time algorithm for counting points on superelliptic curves defined over $\mathbb Q$ that is substantially faster than previous approaches. Our algorithm takes as input a…

We propose and discuss how basic notions (quadratic modules, positive elements, semialgebraic sets, Archimedean orderings) and results (Positivstellensaetze) from real algebraic geometry can be generalized to noncommutative $*$-algebras. A…

One of the difficulties in doing noncommutative projective geometry via explicitly presented graded algebras is that it is usually quite difficult to show flatness, as the Hilbert series is uncomputable in general. If the algebra has a…

We propose a generalization of tropical curves by dropping the rationality and integrality requirements while preserving the balancing condition. An interpretation of such curves as critical points of a certain quadratic functional allows…

As a means to better understanding manifolds with positive curvature, there has been much recent interest in the study of non-negatively curved manifolds which contain points at which all 2-planes have positive curvature. We show that there…

Let $X$ be a smooth projective curve of genus $g$, defined over an algebraically closed field $k$, and let $G$ be a connected reductive group over $k$. We say that a $G$-torsor is essentially finite if it admits a reduction to a finite…