42 research outputs found
Recognizing hyperelliptic graphs in polynomial time
Recently, a new set of multigraph parameters was defined, called
"gonalities". Gonality bears some similarity to treewidth, and is a relevant
graph parameter for problems in number theory and multigraph algorithms.
Multigraphs of gonality 1 are trees. We consider so-called "hyperelliptic
graphs" (multigraphs of gonality 2) and provide a safe and complete sets of
reduction rules for such multigraphs, showing that for three of the flavors of
gonality, we can recognize hyperelliptic graphs in O(n log n+m) time, where n
is the number of vertices and m the number of edges of the multigraph.Comment: 33 pages, 8 figure
Graphs of gonality three
In 2013, Chan classified all metric hyperelliptic graphs, proving that
divisorial gonality and geometric gonality are equivalent in the hyperelliptic
case. We show that such a classification extends to combinatorial graphs of
divisorial gonality three, under certain edge- and vertex-connectivity
assumptions. We also give a construction for graphs of divisorial gonality
three, and provide conditions for determining when a graph is not of divisorial
gonality three.Comment: 19 pages, 13 figures; corrected statements of Theorems 1.2 and 4.1,
as well as material in Section
On computing Belyi maps
We survey methods to compute three-point branched covers of the projective
line, also known as Belyi maps. These methods include a direct approach,
involving the solution of a system of polynomial equations, as well as complex
analytic methods, modular forms methods, and p-adic methods. Along the way, we
pose several questions and provide numerous examples.Comment: 57 pages, 3 figures, extensive bibliography; English and French
abstract; revised according to referee's suggestion
Computing graph gonality is hard
There are several notions of gonality for graphs. The divisorial gonality
dgon(G) of a graph G is the smallest degree of a divisor of positive rank in
the sense of Baker-Norine. The stable gonality sgon(G) of a graph G is the
minimum degree of a finite harmonic morphism from a refinement of G to a tree,
as defined by Cornelissen, Kato and Kool. We show that computing dgon(G) and
sgon(G) are NP-hard by a reduction from the maximum independent set problem and
the vertex cover problem, respectively. Both constructions show that computing
gonality is moreover APX-hard.Comment: The previous version only dealt with hardness of the divisorial
gonality. The current version also shows hardness of stable gonality and
discusses the relation between the two graph parameter
Stable divisorial gonality is in NP
Divisorial gonality and stable divisorial gonality are graph parameters,
which have an origin in algebraic geometry. Divisorial gonality of a connected
graph can be defined with help of a chip firing game on . The stable
divisorial gonality of is the minimum divisorial gonality over all
subdivisions of edges of .
In this paper we prove that deciding whether a given connected graph has
stable divisorial gonality at most a given integer belongs to the class NP.
Combined with the result that (stable) divisorial gonality is NP-hard by
Gijswijt, we obtain that stable divisorial gonality is NP-complete. The proof
consist of a partial certificate that can be verified by solving an Integer
Linear Programming instance. As a corollary, we have that the number of
subdivisions needed for minimum stable divisorial gonality of a graph with
vertices is bounded by for a polynomial
Theta surfaces
A theta surface in affine 3-space is the zero set of a Riemann theta function
in genus 3. This includes surfaces arising from special plane quartics that are
singular or reducible. Lie and Poincar\'e showed that theta surfaces are
precisely the surfaces of double translation, i.e. obtained as the Minkowski
sum of two space curves in two different ways. These curves are parametrized by
abelian integrals, so they are usually not algebraic. This paper offers a new
view on this classical topic through the lens of computation. We present
practical tools for passing between quartic curves and their theta surfaces,
and we develop the numerical algebraic geometry of degenerations of theta
functions.Comment: 28 pages, 8 figures. v2: exposition improved, new references added,
better numerical experiments. To appear in Vietnam J. Math. for J\"urgen
Jost's 65th birthda
On the zero-dispersion limit of the Benjamin-Ono Cauchy problem for positive initial data
We study the Cauchy initial-value problem for the Benjamin-Ono equation in
the zero-disperion limit, and we establish the existence of this limit in a
certain weak sense by developing an appropriate analogue of the method invented
by Lax and Levermore to analyze the corresponding limit for the Korteweg-de
Vries equation.Comment: 54 pages, 11 figure