R. Crew conjectured that every overconvergent F-isocrystal over k((t)) (k a field of positive characteristic) is quasi-unipotent (equivalently, potentially semistable), and so has ``generic'' and ``special'' Newton polygons. It is easy to construct a Newton polygon for an arbitrary overconvergent F-crystal that coincides with the generic Newton polygon for potentially semistable crystals. We give an analogous construction for the special Newton polygon, by showing that the F-crystal can be trivialized over a large auxiliary ring. In a subsequent preprint, we use this construction to prove the aforementioned conjecture.
We consider p-divisible groups (also called Barsotti-Tate groups) in characteristic p, their deformations, and we draw some conclusions. For such a group we can define its Newton polygon (abbreviated NP). This is invariant under isogeny. For an abelian variety (in characteristic p) the Newton polygon of its p-divisible group is ``symmetric''. In 1963 Manin conjectured that conversely any symmetric Newton polygon is ``algebroid''; i.e., it is the Newton polygon of an abelian variety. This conjecture was shown to be true and was proved with the help of the ``Honda-Serre-Tate theory''. We give an…
Let $A$ be an abelian variety of Mumford's type. This paper determines all possible Newton polygons of $A$ in char $p$. This work generalizes a result of R. Noot.
In this paper we characterize, in two different ways, the Newton polygons which are jacobian Newton polygons of a branch. These characterizations give in particular combinatorial criteria of irreducibility for complex series in two variables and necessary conditions which a complex curve has to satisfy in order to be the discriminant of a complex plane branch.
Let $C$ be a generic complex plane plane curve with a given Newton polygon $P$. We compute the number of its inflection points and bitangents (equivalently, the number of singularities of the projectively dual curve $C^\vee$). We also prove that $C^\vee$ has no singularities other than nodes and cusps for large enough polygons $P$.
We prove constancy of Newton polygons of all convergent $F$-isocrystals on Abelian varieties over finite fields. Applying the constancy, we prove the isotriviality of projective smooth families of curves over Abelian varieties. We also study the isotriviality over simply connected projective smooth varieties.
Adapting the Newton-Puiseux Polygon process to nonlinear q-difference equations of any order and degree, we compute their power series solutions, study the properties of the set of exponents of the solutions and give a bound for their $q-$Gevrey order in terms of the order of the original equation.
The Newton--Okounkov body of a big divisor D on a smooth surface is a numerical invariant in the form of a convex polygon. We study the geometric significance of the shape of Newton--Okounkov polygons of ample divisors, showing that they share several important properties of Newton polygons on toric surfaces. In concrete terms, sides of the polygon are associated to some particular irreducible curves, and their lengths are determined by the intersection numbers of these curves with D. As a consequence of our description we determine the numbers k such that D admits some k-gon as a Newton--Okou…
A p-divisible group over a base scheme in characteristic p in general does not admit a slope filtration. Let X be a p-divisible group with constant Newton polygon over a normal noetherian scheme S; we prove that there exists an isogeny from X to Y such that Y admits a slope filtration. In case S is regular this was proved by N. Katz for dim(S) = 1 and by T. Zink for dim(S) > 0. We give an example of a p-divisible group over a non-normal base which does not admit an isogeny to a p-divisible group with a slope filtration.
In this paper, we shall precise the asymptotic behaviour of Newton polygons of $L$ functions associated to character sums, coming from some $n$ variable Laurent polynomials. In order to do this, we use the free sum on convex polytopes. This operation allows the determination of the limit of generic Newton polygons for the sum $Δ=Δ_1\oplus Δ_2$ when we know the limit of generic Newton polygons for each factor. To our knowledge, these are the first results concerning the asymptotic behaviour of Newton polygons for multivariable polynomials.
The Newton polygon of the implicit equation of a rational plane curve is explicitly determined by the multiplicities of any of its parametrizations. We give an intersection-theoretical proof of this fact based on a refinement of the Kushnirenko-Bernstein theorem. We apply this result to the determination of the Newton polygon of a curve parameterized by generic Laurent polynomials or by generic rational functions, with explicit genericity conditions. We also show that the variety of rational curves with given Newton polygon is unirational and we compute its dimension. As a consequence, we obta…
We consider polynomially and rationally parameterized curves, where the polynomials in the parameterization have fixed supports and generic coefficients. We apply sparse (or toric) elimination theory in order to determine the vertex representation of its implicit polygon, i.e. of the implicit equation's Newton polygon. In particular, we consider mixed subdivisions of the input Newton polygons and regular triangulations of point sets defined by Cayley's trick. We distinguish polynomial and rational parameterizations, where the latter may have the same or different denominators; the implicit pol…
In this paper we compute the first vertex of a generic Newton polygon in some special cases, and the corresponding Hasse polynomial. This allows us to show the nonexistence of $p$-cyclic coverings of the projective line in characteristic $p$ with supersingular jacobian for some (infinite families of) genera.
We study the $p$-adic absolute value of the roots of the $L$-functions associated to certain twisted character sums, and additive character sums associated to polynomials $P(x^d)$, when $P$ varies among the space of polynomial of fixed degree $e$ over a finite field of characteristic $p$. For sufficiently large $p$, we determine in both cases generic Newton polygons for these $L$-functions, which is a lower bound for the Newton polygons, and the set of polynomials of degree $e$ for which this generic polygon is attained. In the case of twisted sums, we show that the lower polygon defined in \c…
In this work, we discuss graph like image of curves under moment maps and their relation with the Newton polygon of the curve, which has applications to Lagrangian torus fibration of Calabi-Yau manifolds.
Suppose $X$ is a smooth projective connected curve defined over an algebraically closed field $k$ of characteristic $p>0$ and $B \subset X(k)$ is a finite, possibly empty, set of points. The Newton polygon of a degree $p$ Galois cover of $X$ with branch locus $B$ depends on the ramification invariants of the cover. When $X$ is ordinary, for every possible set of branch points and ramification invariants, we prove that there exists such a cover whose Newton polygon is minimal or close to minimal.
In this paper, we determine the bifurcation set of a real polynomial function of two variables for non-degenerate case in the sense of Newton polygons by using a toric compactification. We also count the number of singular phenomena at infinity, called "cleaving" and "vanishing" in the same setting. Finally, we give an upper bound of the number of elements in the bifurcation set in terms of its Newton polygon. To obtain the upper bound, we apply toric modifications to the singularities at infinity successively.
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.
We study the variation of the convergence Newton polygon of a differential equation along a smooth Berkovich curve over a non-archimedean complete valued field of characteristic 0. Relying on work of the second author who investigated its properties on affinoid domains of the affine line, we prove that its slopes give rise to continuous functions that factorize by the retraction through a locally finite subgraph of the curve.
Let I be an arbitrary ideal in C[[x,y]]. We use the Newton algorithm to compute by induction the motivic zeta function of the ideal, yielding only few poles, associated to the faces of the successive Newton polygons. We associate a minimal Newton tree to I, related to using good coordinates in the Newton algorithm, and show that it has a conceptual geometric interpretation in terms of the log canonical model of I. We also compute the log canonical threshold from a Newton polygon and strengthen Corti's inequalities.
There are open questions about which Newton polygons and Ekedahl-Oort types occur for Jacobians of smooth curves of genus $g$ in positive characteristic $p$. In this chapter, I survey the current state of knowledge about these questions. I include a new result, joint with Karemaker, which verifies, for each odd prime $p$, that there exist supersingular curves of genus $g$ defined over $\overline{\mathbb F}_p$ for infinitely many new values of $g$. I sketch a proof of Faber and Van der Geer's theorem about the geometry of the $p$-rank stratification of the moduli space of curves. The chapter en…
The isomorphism number (resp. isogeny cutoff) of a p-divisible group D over an algebraically closed field is the least positive integer m such that D[p^m] determines D up to isomorphism (resp. up to isogeny). We show that these invariants are lower semicontinuous in families of p-divisible groups of constant Newton polygon. Thus they allow refinements of Newton polygon strata. In each isogeny class of p-divisible groups, we determine the maximal value of isogeny cutoffs and give an upper bound for isomorphism numbers, which is shown to be optimal in the isoclinic case. In particular, the latte…
We survey results and open questions about the $p$-ranks and Newton polygons of Jacobians of curves in positive characteristic $p$. We prove some geometric results about the $p$-rank stratification of the moduli space of (hyperelliptic) curves. For example, if $0 \leq f \leq g-1$, we prove that every component of the $p$-rank $f+1$ stratum of ${\mathcal M}_g$ contains a component of the $p$-rank $f$ stratum in its closure. We prove that the $p$-rank $f$ stratum of $\overline{\mathcal M}_g$ is connected. For all primes $p$ and all $g \geq 4$, we demonstrate the existence of a Jacobian of a smoo…
This paper is dedicated to the description of the poles of the Igusa local zeta functions $Z(s,f,v)$ when $f(x,y)$ satisfies a new non degeneracy condition, that we have called arithmetic non degeneracy. More precisely, we attach to each polynomial $f(x,y)$, a collection of convex sets $Γ^{A}$ called the arithmetic Newton polygon of $f(x,y)$, and introduce the notion of arithmetic non degeneracy with respect to $Γ^{A}(f)$. The set of degenerate polynomials, with respect to a fixed geometric Newton polygon, contains an open subset, for the Zariski topology, formed by non degenerate polynomials …
In this paper we consider the Newton polygons of $L$-functions coming from additive exponential sums associated to a polynomial over a finite field $\F_q$. These polygons define a stratification of the space of polynomials of fixed degree. We determine the open stratum: we give the generic Newton polygon for polynomials of degree $d\geq 2$ when the characteristic $p$ is greater than 3d, and the Hasse polynomial, i.e. the equation defining the hypersurface complementary to the open stratum.
We consider a continuous family $(f_s)$, $s\in[0,1]$ of complex polynomials in two variables with isolated singularities, that are Newton non-degenerate. We suppose that the Euler characteristic of a generic fiber is constant (or equivalently the sum of the affine Milnor number and the Milnor number at infinity $μ(s)+λ(s)$ is constant). We firstly prove that the set of critical values at infinity depends continuously on $s$, and secondly that the degree of the $f_s$ is constant (up to an algebraic automorphism of $\Cc^2$).
Let a planar algebraic curve $C$ be defined over a valuation field by an equation $F(x,y)=0$. Valuations of the coefficients of $F$ define a subdivision of the Newton polygon $Δ$ of the curve $C$. If a given point $p$ is of multiplicity $m$ for $C$, then the coefficients of $F$ are subject to certain linear constraints. These constraints can be visualized on the above subdivision of $Δ$. Namely, we find a distinguished collection of faces of the above subdivision, with total area at least $\frac{3}{8}m^2$. In a sense, the union of these faces in "the region of influence" of the singular point …
Let P(x) be a one-variable Laurent polynomial of degree (d_1,d_2) over a finite field of characteristic p. For any fixed positive integer s not divisible by p, we prove that the (normalized) p-adic Newton polygon of the L-functions of exponential sums of P(x^s) has a tight lower bound which we call `Hodge-Stickelberger polygon', depending only on d_1,d_2,s, and (p mod s). This Hodge-Stickelberger polygon is a weighted convolution of a `Hodge polygon' for L-function of exponential sum of P(x) and the `Newton polygon' for L-function of exponential sum of x^s (given by the classical Stickelberger…
Let p be a prime and let F_pbar be the algebraic closure of the finite field of p elements. Let f(x) be any one variable rational function over F_pbar with n poles of orders d_1, ...,d_n. Suppose p is coprime to d_i for every i. We prove that there exists a Hodge polygon, depending only on d_i's, which is a lower bound to the Newton polygon of L functions of exponential sums of f(x). Moreover, we show that these two polygons coincide if p=1 mod d_i for every i=1,...,n. As a corollary, we obtain a tight lower bound of Newton polygon of Artin-Schreier curve.
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.
In this paper we use the connections between tropical algebraic geometry and rigid analytic geometry in order to prove two main results. We use tropical methods to prove a theorem about the Newton polygon for convergent power series in several variables: if f_1,...,f_n are n convergent power series in n variables with coefficients in a non-Archimedean field K, we give a formula for the valuations and multiplicities of the common zeros of f_1,...,f_n. We use rigid-analytic methods to show that stable complete intersections of tropical hypersurfaces compute algebraic multiplicities even when the…
This paper generalizes the classical theory of Newton polygons from the case of general linear groups to the case of split reductive groups. It also gives a root-theoretic formula for dimensions of Newton strata in the adjoint quotients of reductive groups.
Let $C$ be a smooth projective curve in $\mathbb{P}^1\times \mathbb{P}^1$ of genus $g\neq 4$, and assume that it is birationally equivalent to a curve defined by a Laurent polynomial that is non-degenerate with respect to its Newton polygon $Δ$. Then we show that the convex hull $Δ^{(1)}$ of the interior lattice points of $Δ$ is a standard rectangle, up to a unimodular transformation. Our main auxiliary result, which we believe to be interesting in its own right, is that the first scrollar Betti numbers of $Δ$-non-degenerate curves are encoded in the combinatorics of $Δ^{(1)}$, if $Δ$ satisfie…
Let K be a local field of characteristic zero, O its ring of integers and F(x) a monic irreducible polynomial with coefficients in O. K. Okutsu attached to F(x) certain primitive divisor polynomials F_1(x),..., F_r(x), that are specially close to F(x) with respect to their degree. In this paper we characterize the Okutsu families [F_1,..., F_r] in terms of certain Newton polygons of higher order, and we derive some applications: closed formulas for certain Okutsu invariants, the discovery of new Okutsu invariants, or the construction of Montes approximations to F(x); these are monic irreducibl…
This is the LaTeX version of the handwritten notes of a lecture at the Kyoto Singularities Symposium held at the RIMS in April 1978. It presents the relationship of various invariants of isolated singularities of complex analytic hypersurfaces with the jacobian Newton polygon built from the polar curves. An appendix presents features of the semiring of Newton polygons, because the product of Jacobian Newton polygons corresponds to the Thom-Sebastiani sum of hypersurfaces. Analogies between mixed multiplicities and mixed volumes are noticed. They have been explained since.
The L-function of a non-degenerate twisted Witt extension is proved to be a polynomial. Its Newton polygon is proved to lie above the Hodge polygon of that extension. And the Newton polygons of the Gauss-Heilbronn sums are explicitly determined, generalizing the Stickelberger theorem.
Bi-Lipschitz invariance of Newton polygons along gradient canyons
Air Force Institute of Technology（空军理工学院）；School of Mathematics and Statistics, University of Sydney（悉尼大学数学与统计学院）；Mathématiques, UMR 8524 CNRS, Université de Lille（法国里尔大学CNRS 8524数学实验室）
Consider an $F$-crystal over a noetherian scheme $S$. De Jong--Oort's purity theorem states that the associated Newton polygons over all points of $S$ are constant if this is true outside a subset of codimension bigger than 1. In this paper we show an improvement of the theorem, which says that the Newton polygons over all points of $S$ have a common break point if this is true outside a subset of codimension bigger than 1.
The twisted $T$-adic exponential sum associated to $x^{d}+λx$ is studied. If $λ\neq0,$ then an explicit arithmetic polygon is proved to be the Newton polygon of the twisted $C$-function of the T-adic exponential sum. It gives the Newton polygons of the $L$-functions of twisted $p$-power order exponential sums.
In this article, we use Robba's method to give an estimate of the Newton polygon for the L function on torus and we can draw the Newton polygon in some special cases.
The twisted $T$-adic exponential sum associated to a polynomial in one variable is studied. An explicit arithmetic polygon is proved to be the generic Newton polygon of the twisted $C$-function of the T-adic exponential sum. It gives the generic Newton polygon of the twisted $L$-functions of $p^m$-power order exponential sums, for all $m$.
The $T$-adic exponential sum associated to a Laurent polynomial in one variable is studied. An explicit arithmetic polygon is proved to be the generic Newton polygon of the $C$-function of the T-adic exponential sum. It gives the generic Newton polygon of $L$-functions of $p$-power order exponential sums.
Let $k$ be a perfect field and let $C_0:f=0$ be a smooth curve in the torus $\mathbb{G}_{m,k}^2$. Let $\mathbb{T}_Δ$ be the toric variety associated to the Newton polygon of $f$. Extending the toric resolution of $C_0$ on $\mathbb{T}_Δ$, we construct an explicit model over $k$ of the smooth completion of $C_0$. Such a model exists for any smooth projective curve and can be described via a combinatorial algorithm using an iterative construction of Newton polygons.
The $T$-adic exponential sum of a polynomial in one variable is studied. An explicit arithmetic polygon in terms of the highest two exponents of the polynomial is proved to be a lower bound of the Newton polygon of the $C$-function of the T-adic exponential sum. This bound gives lower bounds for the Newton polygon of the $L$-function of exponential sums of $p$-power order.
We investigate the equisingularity question for $1$-parameter deformation families of mixed polynomial functions $f_t(\mathbf{z},\bar{\mathbf{z}})$ from the Newton polygon point of view. We show that if the members $f_t$ of the family satisfy a number of elementary conditions, which can be easily described in terms of the Newton polygon, then the corresponding family of mixed hypersurfaces $f_t^{-1}(0)$ is Whitney equisingular (and hence topologically equisingular) and satisfies the Thom condition.
This survey may be seen as an introduction to the use of toric and tropical geometry in the analysis of plane curve singularities, which are germs $(C,o)$ of complex analytic curves contained in a smooth complex analytic surface $S$. The embedded topological type of such a pair $(S, C)$ is usually defined to be that of the oriented link obtained by intersecting $C$ with a sufficiently small oriented Euclidean sphere centered at the point $o$, defined once a system of local coordinates $(x,y)$ was chosen on the germ $(S,o)$. If one works more generally over an arbitrary algebraically closed fie…
Let $C$ be an algebraic curve defined by a sufficiently generic bivariate Laurent polynomial with given Newton polygon $Δ$. It is classical that the geometric genus of $C$ equals the number of lattice points in the interior of $Δ$. 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 'co…
In 2004 Oort studied the foliation on the space of $p$-divisible groups. In his theory, special leaves called central streams play an important role. It is still meaningful to investigate central streams, for example, there remain a lot of unknown things on the boundaries of central streams. In this paper, we classify the boundary components of the central stream for a Newton polygon consisting of two segments, where one slope is less than $1/2$ and the other slope is greater than $1/2$. Moreover we determine the generic Newton polygon of each boundary component using this classification.
Let $f$ be a Laurent polynomial in two variables, whose Newton polygon strictly contains the origin and whose vertices are primitive lattice points, and let $L_f$ be the minimal-order differential operator that annihilates the period integral of $f$. We prove several results about $f$ and $L_f$ in terms of the Newton polygon of $f$ and the combinatorial operation of *mutation*, in particular we give an in principle complete description of the monodromy of $L_f$ around the origin. Special attention is given to the class of *maximally mutable* Laurent polynomials, which has applications to the c…
We introduce an efficient way, called Newton algorithm, to study arbitrary ideals in C[[x,y]], using a finite succession of Newton polygons. We codify most of the data of the algorithm in a useful combinatorial object, the Newton tree. For instance when the ideal is of finite codimension, invariants like integral closure and Hilbert-Samuel multiplicity were already combinatorially determined in the very special cases of monomial or non degenerate ideals, using the Newton polygon of the ideal. With our approach, we can generalize these results to arbitrary ideals. In particular the Rees valuati…
