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.     

On a conjecture of Wan about limiting Newton polygons

We show that for a monic polynomial $f(x)$ over a number field K containing a global permutation polynomial of degree >1 as its composition factor, the Newton Polygon of $f\mathrm{mod}\mathfrak{p}$ does not converge for $\mathfrak{p}$ passing through all finite places of K. In the rational number field case, our result is the “only if” part of a conjecture of Wan about limiting Newton polygons.     

On polygons circumscribing a closed convex curve

Length and area formulas for closed polygonal curves are derived, as functions of the vertex angles and the distances to the lines containing the sides. Applications of the formulas are made to the class of polygons which circumscribe a given convex curve and have a prescribed sequence of vertex angles. Geometric conditions are given for polygons in the class which have extremal perimeter or area.     

Classification of limiting shapes for isotropic curve flows

A complete classification is given of curves in the plane which contract homothetically when evolved according to a power of their curvature. Applications are given to the limiting behaviour of the flows in various situations.

     

The mean of circumscribed polygons

The mean of circumscribed polygons of a convex body in \Bbb R ²\Bbb R ^2 is again a convex body. The corresponding mapping of convex bodies is an endomorphism, and we characterize the injective case. It is shown that this endomorphism is almost always injective.     

The Poincaré metric and isoperimetric inequalities for hyperbolic polygons

We prove several isoperimetric inequalities for the conformal radius (or equivalently for the Poincaré density) of polygons on the hyperbolic plane. Our results include, as limit cases, the isoperimetric inequality for the conformal radius of Euclidean -gons conjectured by G. Pólya and G. Szegö in 1951 and a similar inequality for the hyperbolic -gons of the maximal hyperbolic area conjectured by J. Hersch. Both conjectures have been proved in previous papers by the third author.

Our approach uses the method based on a special triangulation of polygons and weighted inequalities for the reduced modules of trilaterals developed by A. Yu. Solynin. We also employ the dissymmetrization transformation of V. N. Dubinin. As an important part of our proofs, we obtain monotonicity and convexity results for special combinations of the Euler gamma functions, which appear to have a significant interest in their own right.

     

正多边形对称群的子群

利用Lagrange定理和正多边形对称群的性质,首先对正多边形对称群的子群的性质进行了研究,其次讨论了正多边形对称群的子群的结构,由此完全确定了正多边形对称群的子群,最后应用所得结论求出了正六边形对称群的所有子群.     

The limit shape of convex lattice polygons

It is proved here that, asn→∞, almost all convex (1/n)ℤ²-lattice polygons lying in the square [−1, 1]² are very close to a fixed convex set. This research was partially supported by Hungarian Science Foundation Grants 1907 and 1909.     

The approximation of cauchy singular integrals and their limiting values at the endpoints of the curve of integration

We examine a specific approximating process for the singular integral $$S^* (f;x) \equiv \frac{1}{\pi }\int_{ - 1}^{ + 1} {\frac{{f(t)}}{{\sqrt {1 - l^2 } (t - x)}}} dt( - 1< x< 1)$$ taken in the principal value sense. We study the influence of some local properties of the functionf on the convergence of the approximations. Next, assuming that \(S^* (f;c) \equiv \mathop {\lim }\limits_{x \to c} S^* (f;x)\) , where c is an arbitrary one of the endpoints ?1 and 1, we show that the conditions which guarantee the existence of the limiting values S*(f; c) (c=±1) and, moreover, the convergence of the process at an arbitrary point x∈ (?1, 1) are not always sufficient for convergence of the approximations at the endpoints.     

The structure of near polygons with quads

We develop a structure theory for near polygons with quads. Main results are the existence of sub 2j-gons for 2?j?d and the nonexistence of regular sporadic 2d-gons for d?4 with s>1 and t ₂>1 and t ₃≠t ₂(t ₂+1).     

The non-existence of certain regular generalized polygons

Senior Research Associate at the National Fund for Scientific Research (Belgium)     

Recuttings of polygons

L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 27, No. 2, pp. 79–82, April–June, 1993.     

The cross covariogram of a pair of polygons determines both polygons, with a few exceptions

The cross covariogram g_K,L of two convex sets K and L in is the function which associates to each the volume of K∩(L+x). We prove that when K and L are either convex polygons or planar convex cones, g_K,L determines both K and L, up to a described family of exceptions. These results imply that, when K and L are in these classes, the information provided by the cross covariogram is so rich as to determine not only one unknown body, as required by Matheron's conjecture, but two bodies, with a few classified exceptions.     

The geometry of circumscribing polygons of minimal perimeter

LetC be a closed strictly convex curve in the Euclidean plane, and letC(n) denote the class ofn-sided polygons which circumscribeC. Geometric conditions are given for polygonsP C(n) which have minimum perimeter in the classC(n).     

X-rays of polygons

Various results are given concerning X-rays of polygons in ². It is shown that no finite set of X-rays determines every star-shaped polygon, partially answering a question of S. Skiena. For anyn, there are simple polygons which cannot be verified by any set ofn X-rays. Convex polygons are uniquely determined by X-rays at any two points. Finally, it is proved that given a convex polygon, certain sets of three X-rays will distinguish it from other Lebesgue measurable sets.This work was done at the Istituto Analisi Globale e Applicazioni, Florence, Italy.