Related papers: Eighth problem of the Arnold trivium

We determine the gonality and the Clifford index for curves on a compact smooth toric surface. Moreover, it is shown that their gonality are computed by pencils on the ambient surface. From the geometrical view point, this means that the…

The article is devoted to the development of numerical methods for solving saddle point problems and variational inequalities with simplified requirements for the smoothness conditions of functionals. Recently there were proposed some…

Aronhold's classical result states that a plane quartic can be recovered by the configuration of any Aronhold systems of bitangents, i.e. special 7-tuples of bitangents such that the six points at which any subtriple of bitangents touches…

We clarify the question whether for a smooth curve of polynomials one can choose the roots smoothly and related questions. Applications to perturbation theory of operators are given.

We determine the number of Del Pezzo surfaces of degree 2 over finite fields of odd characteristic with specified action of the Frobenius endomorphism, i.e. we solve the "quantitative inverse Galois problem". As applications we determine…

We compute the class of arithmetic genus two Teichmueller curves in the Picard group of pseudo-Hilbert modular surfaces, distinguished according to their torsion order and spin invariant. As an application, we compute the number of genus…

We study a broad class of polynomial optimization problems whose constraints and objective functions exhibit sparsity patterns. We give two characterizations of the number of critical points to these problems, one as a mixed volume and one…

For a real valued function, a point is critical if its derivatives are zero, and a critical point is a saddle point if it is not a local extrema. In this paper, we study algorithms to find saddle points of general Morse index. Our approach…

The (global) Lipschitz smoothness condition is crucial in establishing the convergence theory for most optimization methods. Unfortunately, most machine learning and signal processing problems are not Lipschitz smooth. This motivates us to…

We give a new construction of the irregular, generalized Lagrangian, surfaces of general type with p_g=5, \chi=2, K^2=8, recently discovered by Chad Schoen. Our approach proves that, if S is a general Schoen surface, its canonical map is a…

We complete the classification of all smooth 4-dimensional Kahler geometries admitting a twistor (conformal Killing-Yano) 2-form invariant under a 2-torus action. We establish that there are six geometrically distinct families, and we…

Recently Arnold's $\St$ and $J^{\pm}$ invariants of generic planar curves have been generalized to the case of generic planar wave fronts. We generalize these invariants to the case of wave fronts on an arbitrary surface $F$. All invariants…

We explore Arnold's $J^+$-invariant of immersions -- planar smooth closed curves with non-vanishing derivative, at most double points and only transverse intersections -- and computation methods like Viro's sum, among others. Only basic…

We consider 2-surfaces arising from the Korteweg de Vries (KdV) equation. The surfaces corresponding to KdV are in a three dimensional Minkowski space. They contain a family of quadratic Weingarten and Willmore-like surfaces. We show that a…

We consider a one-dimensional singularly perturbed 4th order problem with the additional feature of a shift term. An expansion into a smooth term, boundary layers and an inner layer yields a formal solution decomposition, and together with…

Normal surface theory is a central tool in algorithmic three-dimensional topology, and the enumeration of vertex normal surfaces is the computational bottleneck in many important algorithms. However, it is not well understood how the number…

We draw a connection between the affine invariant surface measures constructed by P. Gressman and the boundedness of a certain geometric averaging operator associated to surfaces of codimension $2$ and related to the Fourier Restriction…

A formula expressing a point of order 8 on an elliptic curve, in terms of the roots of the associated cubic polynomial, is given. Doubling such a point yields a point of order 4 distinct from the well-known points of order 4 given in…

Given a closed, oriented surface, possibly with boundary, and a mapping class, we obtain sharp lower bounds on the number of fixed points of a surface symplectomorphism (i.e. area-preserving map) in the given mapping class, both with and…

We give an algorithm that takes a smooth hypersurface over a number field and computes a $p$-adic approximation of the obstruction map on the Tate classes of a finite reduction. This gives an upper bound on the "middle Picard number" of the…