Conjugacy classes of affine automorphisms of and linear automorphisms of ℙ n in the Cremona groups

We describe the conjugacy classes of affine automorphisms in the group Aut(n,) (respectively Bir()) of automorphisms (respectively of birational maps) of . From this we deduce also the classification of conjugacy classes of automorphisms of ℙⁿ in the Cremona group Bir().     

On the automorphisms of some line congruences in ℙ3

The automorphisms of line congruences in ³ are studied via the analysis of the automorphisms of the associated focal loci. This study is applied to a Veronese surface (i.e. to a congruence of chords of a twisted cubic) and to the rational scrolls in the Grassmannian G(1, 3).     

Rationality of the moduli spaces of plane curves of sufficiently large degree

We prove that the moduli space of plane curves of degree d is rational for all sufficiently large d.     

On the bigℳ in the affine scaling algorithm

When we apply the affine scaling algorithm to a linear program, we usually construct an artificial linear program having an interior feasible solution from which the algorithm starts. The artificial linear program involves a positive number called the big. Theoretically, there exists an ^* such that the original problem to be solved is equivalent to the artificial linear program if > ^*. Practically, however, such an ^* is unknown and a safe estimate of is often too large. This paper proposes a method of updating to a suitable value during the iteration of the affine scaling algorithm. As becomes large, the method gives information on infeasibility of the original problem or its dual.This paper is dedicated to Phil Wolfe on the occasion of his 65th birthday.Supported by Grant-in-Aids for Co-Operative Research (03832017) of the Japan Ministry of Education, Science and Culture.     

On a conjecture of the Randić index and the minimum degree of graphs

The Randi? index $R\left(G\right)$ of a graph $G$ is defined by $R\left(G\right)={\sum }_{uv}\frac{1}{\sqrt{d\left(u\right)d\left(v\right)}}$, where $d\left(u\right)$ is the degree of a vertex $u$ and the summation extends over all edges $uv$ of $G$. Delorme et al. (2002)  [6] put forward a conjecture concerning the minimum Randi? index among all$n$-vertex connected graphs with the minimum degree at least $k$. In this work, we show that the conjecture is true given the graph contains $k$ vertices of degree $n?1$. Further, it is true among $k$-trees.     

On the index of contact and multiplicities¶for bigraded rings

We prove that the index of contact p(Z,S)(c) of a set Z with a submanifold S at a point c∈ reg(Z∩S) at which Z is normal pseudo-flat along Z∩S coincides with the Samuel multiplicity of the associated graded ring and give various methods of computing this invariant at such points. Received: 12 July 2001     

On the transposed Rao’s flag-transitive plane of order 49

Rao’s flag-transitive plane π of order 49 and π ^t , the plane obtained by transposing matrices of a representative set of π, has been studied. It is shown that π ^t is flag-transitive, π ^t is not isomorphic to π, and π ^t is obtained from π by replacement of a net of degree 25. Further, (1) the flag-transitive planes associated with 1-spread sets S _2b and S _2a in the classified list of translation planes of order 49 enumerated by Mathon et al, are respectively isomorphic to π and π ^t (2) The flag-transitive planes associated with the 1-spread sets of 0an* in the classified list of translation planes of order 49 enumerated by Charnes et al are isomorphic to π and π ^t in some order.     

On Bernstein–Walsh-type lemmas in regions of the complex plane

Let G ì \mathbb C G \subset {\mathbb C} be a finite region bounded by a Jordan curve L: = ?G L: = \partial G , let W: = \textext[`(G)] \Omega : = {\text{ext}}\bar{G} (with respect to [`(\mathbb C)] {\overline {\mathbb C}} ), $ \Delta : = \left\{ {z:\left| z \right| > 1} \right\} $ \Delta : = \left\{ {z:\left| z \right| > 1} \right\} , and let w = F(z) w = \Phi (z) be a univalent conformal mapping of Ω onto Δ normalized by $ \Phi \left( \infty \right) = \infty, \;\Phi '\left( \infty \right) > 0 $ \Phi \left( \infty \right) = \infty, \;\Phi '\left( \infty \right) > 0 . By A _p (G); p > 0; we denote a class of functions f analytic in G and satisfying the condition

On the number of points of bounded height¶on arithmetic projective spaces

Let K be a number field and its ring of integers. Let be a Hermitian vector bundle over . In the first part of this paper we estimate the number of points of bounded height in (generalizing a result by Schanuel). We give then some applications: we estimate the number of hyperplanes and hypersurfaces of degree d>1 in of bounded height and containing a fixed linear subvariety and we estimate the number of points of height, with respect to the anticanonical line bundle, less then T (when T goes to infinity) of ℙ ^N _K blown up at a linear subspace of codimension two. Received: 20 February 1998 / Revised version: 9 November 1998     

On the (n − 2)-transversals of n convex subsets of the plane

In this paper we consider families of distinct ovals in the plane, with the property that certain subfamilies have stabbing lines (transversals). Our main result says that if any k member of the family can be stabbed by a line avoiding all the other ovals and k is large enough, then the family consists of at most k+1 ovals. For any n4 we show a family of n ovals, whose n–2 element subfamilies have, but the n–1 element subfamilies do not have, transversals.     

On the degree of approximation by the operators of de la Vallée Poussin

In this paper we study the degree of approximation of functionsf inC ₂ andC ₂ ¹ by the operatorsV _n ofde la Vallée Poussin. The quality of approximation is measured in terms of the modulus of continuity off andf respectively. Forn so-called exact constants of approximation are determined. Furthermore, the asymptotic behaviour of these constants is investigated asn.     

On the base point freeness of adjoint bundles¶on normal surfaces

We introduce some invariants of singularities which represent the anti-freeness of the adjoint linear systems. The invariants indicate that if either the singularities or the boundaries are worse then the adjoint linear systems are much global generative. Using these invariants, we prove effective global generation of adjoint linear systems on normal log surfaces. Received: 14 May 1999/ Revised version: 2 August 1999     

On the enumerative nature of Gomory’s dual cutting plane method

For 30 years after their invention half a century ago, cutting planes for integer programs have been an object of theoretical investigations that had no apparent practical use. When they finally proved their practical usefulness in the late eighties, that happened in the framework of branch and bound procedures, as an auxiliary tool meant to reduce the number of enumerated nodes. To this day, pure cutting plane methods alone have poor convergence properties and are typically not used in practice. Our reason for studying them is our belief that these negative properties can be understood and thus remedied only based on a thorough investigation of such procedures in their pure form. In this paper, the second in a sequence, we address some important issues arising when designing a computationally sound pure cutting plane method. We analyze the dual cutting plane procedure proposed by Gomory in 1958, which is the first (and most famous) convergent cutting plane method for integer linear programming. We focus on the enumerative nature of this method as evidenced by the relative computational success of its lexicographic version (as documented in our previous paper on the subject), and we propose new versions of Gomory’s cutting plane procedure with an improved performance. In particular, the new versions are based on enumerative schemes that treat the objective function implicitly, and redefine the lexicographic order on the fly to mimic a sound branching strategy. Preliminary computational results are reported.     

Minimum degree of the difference of two polynomials over {\mathbb Q}, and weighted plane trees

A weighted bicolored plane tree (or just tree for short) is a bicolored plane tree whose edges are endowed with positive integral weights. The degree of a vertex is defined as the sum of the weights of the edges incident to this vertex. Using the theory of dessins d’enfants, which studies the action of the absolute Galois group on graphs embedded into Riemann surfaces, we show that a weighted plane tree is a graphical representation of a pair of coprime polynomials \(P,Q\in {\mathbb C}\,[x]\) such that: (a)  \(\deg P = \deg Q\) , and \(P\) and \(Q\) have the same leading coefficient; (b) the multiplicities of the roots of  \(P\) (respectively, of  \(Q\) ) are equal to the degrees of the black (respectively, white) vertices of the corresponding tree; (c) the degree of the difference \(P-Q\) attains the minimum which is possible for the given multiplicities of the roots of \(P\)  and  \(Q\) . Moreover, if a tree in question is uniquely determined by the set of its black and white vertex degrees (we call such trees unitrees), then the corresponding polynomials are defined over \({\mathbb Q}\) . The pairs of polynomials \(P,Q\) such that the degree of the difference \(P-Q\) attains the minimum, and especially those defined over \({\mathbb Q}\) , are related to some important questions of number theory. Dozens of papers, from 1965 (Birch et al. in Norske Vid Selsk Forh 38:65–69, 1965) to 2010 (Beukers and Stewart in J Number Theory 130:660–679, 2010), were dedicated to their study. The main result of this paper is a complete classification of the unitrees, which provides us with the most massive class of such pairs defined over  \({\mathbb Q}\) . We also study combinatorial invariants of the Galois action on trees, as well as on the corresponding polynomial pairs, which permit us to find yet more examples defined over  \({\mathbb Q}\) . In a subsequent paper, we compute the polynomials \(P,Q\) corresponding to all the unitrees.     

On the unit group and the class number¶of certain composita of two real quadratic fields

The purpose of this paper is to exhibit a new family of real bicyclic biquadratic fields K for which we can write the Hasse unit index of the group generated by the units of the three quadratic subfields in the unit group E _K of K. As a byproduct, one can explicitly relate the class number of K with the product of the class numbers of the three quadratic subfields. Received: 25 July 2000 / Revised version: 12 December 2000