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Between his arrival in Frankfurt in 1922 and and his proof of his famous finiteness theorem for integral points in 1929, Siegel had no publications. He did, however, write a letter to Mordell in 1926 in which he explained a proof of the finiteness of integral points on hyperelliptic curves. Recognizing the importance of this argument (and Siegel's views on publication), Mordell sent the relevant extract to be published under the pseudonym "X".
The purpose of this note is to explain how to optimize Siegel's 1926 technique to obtain the following bound. Let K be a number field, S a finite set of places of K, and f∈oK,S[t] monic of degree d≥5 with discriminant Δf∈o×K,S. Then: #|{(x,y):x,y∈oK,S,y2=f(x)}|≤2rankJac(Cf)(K)⋅O(1)d3⋅([K:Q]+#|S|).
This improves bounds of Evertse-Silverman and Bombieri-Gubler from 1986 and 2006, respectively.
The main point underlying our improvement is that, informally speaking, we insist on "executing the descents in the presence of only one root (and not three) until the last possible moment".
Therapy evasion – and subsequent disease progression – is a major challenge in current oncology. An important role in this context seems to be played by various forms of cancer cell dormancy. For example, therapy-induced dormancy, over short timescales, can create serious obstacles to aggressive treatment approaches such as chemotherapy, and long-term dormancy may lead to relapses and metastases even many years after an initially successful treatment. The underlying dormancy-related mechanisms are complex and highly diverse, so that the analysis even of basic patterns of the population-level consequences of dormancy requires abstraction and idealization, as well as the identification of the relevant specific scenarios.
In this paper, we focus on a situation in which individual cancer cells may switch into and out of a dormant state both spontaneously as well as in response to treatment, and over relatively short time-spans. We introduce a mathematical ‘toy model’, based on stochastic agent-based interactions, for the dynamics of cancer cell populations involving individual short-term dormancy, and allow for a range of (multi-drug) therapy protocols. Our analysis shows that in our idealized model, even a small initial population of dormant cells can lead to therapy failure under classical (and in the absence of dormancy successful) single-drug treatments. We further investigate the effectiveness of several multidrug regimes (manipulating dormant cancer cells in specific ways) and provide some basic rules for the design of (multi-)drug treatment protocols depending on the types and parameters of dormancy mechanisms present in the population.
n this paper we study invasion probabilities and invasion times of cooperative parasites spreading in spatially structured host populations. The spatial structure of the host population is given by a random geometric graph on [0,1]n, n∈N, with a Poisson(N)-distributed number of vertices and in which vertices are connected over an edge when they have a distance of at most rN∈Θ(Nβ−1n) for some 0<β<1 and N→∞. At a host infection many parasites are generated and parasites move along edges to neighbouring hosts. We assume that parasites have to cooperate to infect hosts, in the sense that at least two parasites need to attack a host simultaneously. We find lower and upper bounds on the invasion probability of the parasites in terms of survival probabilities of branching processes with cooperation. Furthermore, we characterize the asymptotic invasion time.
An important ingredient of the proofs is a comparison with infection dynamics of cooperative parasites in host populations structured according to a complete graph, i.e. in well-mixed host populations. For these infection processes we can show that invasion probabilities are asymptotically equal to survival probabilities of branching processes with cooperation.
Furthermore, we build in the proofs on techniques developed in [BP22], where an analogous invasion process has been studied for host populations structured according to a configuration model.
We substantiate our results with simulations.
Using limit linear series on chains of curves, we show that closures of certain Brill-Noether loci contain a product of pointed Brill-Noether loci of small codimension. As a result, we obtain new non-containments of Brill-Noether loci, in particular that dimensionally expected non-containments hold for expected maximal Brill-Noether loci. Using these degenerations, we also give a new proof that Brill-Noether loci with expected codimension −ρ≤⌈g/2⌉ have a component of the expected dimension. Additionally, we obtain new non-containments of Brill-Noether loci by considering the locus of the source curves of unramified double covers.
For genus g=2i≥4 and the length g−1 partition μ=(4,2,…,2,−2,…,−2) of 0, we compute the first coefficients of the class of D¯¯¯¯(μ) in PicQ(R¯¯¯¯g), where D(μ) is the divisor consisting of pairs [C,η]∈Rg with η≅OC(2x1+x2+⋯+xi−1−xi−⋯−x2i−1) for some points x1,…,x2i−1 on C. We further provide several enumerative results that will be used for this computation.
For genus g=2i≥4 and the length g−1 partition μ=(4,2,…,2,−2,…,−2) of 0, we compute the first coefficients of the class of D¯¯¯¯(μ) in PicQ(R¯¯¯¯g), where D(μ) is the divisor consisting of pairs [C,η]∈Rg with η≅OC(2x1+x2+⋯+xi−1−xi−⋯−x2i−1) for some points x1,…,x2i−1 on C. We further provide several enumerative results that will be used for this computation.
We prove that the projectivized strata of differentials are not contained in pointed Brill-Noether divisors, with only a few exceptions. For a generic element in a stratum of differentials, we show that many of the associated pointed Brill-Noether loci are of expected dimension. We use our results to study the Auel-Haburcak Conjecture: We obtain new non-containments between maximal Brill-Noether loci in Mg. Our results regarding quadratic differentials imply that the quadratic strata in genus 6 are uniruled.
For genus g=r(r+1)2+1, we prove that via the forgetful map, the universal Prym-Brill-Noether locus Rrg has a unique irreducible component dominating the moduli space Rg of Prym curves.
For genus g=2i≥4 and the length g−1 partition μ=(4,2,…,2,−2,…,−2) of 0, we compute the first coefficients of the class of D¯¯¯¯(μ) in PicQ(R¯¯¯¯g), where D(μ) is the divisor consisting of pairs [C,η]∈Rg with η≅OC(2x1+x2+⋯+xi−1−xi−⋯−x2i−1) for some points x1,…,x2i−1 on C. We further provide several enumerative results that will be used for this computation.