The lattice size of a lattice polygon

We give upper bounds on the minimal degree of a model in $\mathbb{P}^2$ and the minimal bidegree of a model in $\mathbb{P}^1 \times \mathbb{P}^1$ of the curve defined by a given Laurent polynomial, in terms of the combinatorics of the Newton polygon of the latter. We prove in various cases that this bound is sharp as soon as the polynomial is sufficiently generic with respect to its Newton polygon

Newton polygons and curve gonalities

We give a combinatorial upper bound for the gonality of a curve that is defined by a bivariate Laurent polynomial with given Newton polygon. We conjecture that this bound is generically attained, and provide proofs in a considerable number of special cases. One proof technique uses recent work of M. Baker on linear systems on graphs, by means of which we reduce our conjecture to a purely combinatorial statement.Comment: 29 pages, 18 figures; erratum at the end of the articl

The power of creative thinking in situations of uncertainties: the almost impossible task of protecting critical infrastructures

A good and scientific analysis starts with a closer look at the conceptualisation at hand. The definition of CIP is not easy because of its wide range. This paper examines infrastructures that are critical and need protection. Each word entails a specific connotation and is characterized by several components

Linear pencils encoded in the Newton polygon

Let $C$ be an algebraic curve defined by a sufficiently generic bivariate Laurent polynomial with given Newton polygon $\Delta$. It is classical that the geometric genus of $C$ equals the number of lattice points in the interior of $\Delta$. In this paper we give similar combinatorial interpretations for the gonality, the Clifford index and the Clifford dimension, by removing a technical assumption from a recent result of Kawaguchi. More generally, the method shows that apart from certain well-understood exceptions, every base-point free pencil whose degree equals or slightly exceeds the gonality is 'combinatorial', in the sense that it corresponds to projecting $C$ along a lattice direction. We then give an interpretation for the scrollar invariants associated to a combinatorial pencil, and show how one can tell whether the pencil is complete or not. Among the applications, we find that every smooth projective curve admits at most one Weierstrass semi-group of embedding dimension $2$, and that if a non-hyperelliptic smooth projective curve $C$ of genus $g \geq 2$ can be embedded in the $n$th Hirzebruch surface $\mathcal{H}_n$, then $n$ is actually an invariant of $C$.Comment: This covers and extends sections 1 to 3.4 of our previously posted article "On the intrinsicness of the Newton polygon" (arXiv:1304.4997), which will eventually become obsolete. arXiv admin note: text overlap with arXiv:1304.499

The power of creative thinking in situations of uncertainties: the almost impossible task of protecting critical infrastructures

A good and scientific analysis starts with a closer look at the conceptualisation at hand. The definition of CIP is not easy because of its wide range. This paper examines infrastructures that are critical and need protection. Each word entails a specific connotation and is characterized by several components

The analysis of vertex modified lattice rules in a non-periodic Sobolev space

In a series of papers, in 1993, 1994 & 1996, Sloan & Niederreiter introduced a modification of lattice rules for non-periodic functions, called "vertex modified lattice rules"', and a particular breed called "optimal vertex modified lattice rules". In the 1994 paper, Niederreiter & Sloan concentrate explicitly on Fibonacci lattice rules, which are a particular good choice of 2-dimensional lattice rules. Error bounds in this series of papers were given related to the star discrepancy. In this paper we pose the problem in terms of the so-called unanchored Sobolev space, which is a reproducing kernel Hilbert space often studied nowadays in which functions have $L_2$-integrable mixed first derivatives. It is known constructively that randomly shifted lattice rules, as well as deterministic tent-transformed lattice rules and deterministic fully symmetrized lattice rules can achieve close to $O(N^{-1})$ convergence in this space, see Sloan, Kuo & Joe (2002) and Dick, Nuyens & Pillichshammer (2014) respectively. We derive a break down of the worst-case error of vertex modified lattice rules in the unanchored Sobolev space in terms of the worst-case error in a Korobov space, a multilinear space and some additional "mixture term". For the 1-dimensional case this worst-case error is obvious and gives an explicit expression for the trapezoidal rule. In the 2-dimensional case this mixture term also takes on an explicit form for which we derive upper and lower bounds. For this case we prove that there exist lattice rules with a nice worst-case error bound with the additional mixture term of the form $N^{-1} \log^2(N)$.Comment: The data from the table and the figure can be downloaded as the file "wce.csv" (comma separated values