Abstract. The central configurations of the Newtonian n-body problem are configurations of n point bodies, each with a positive mass, such that the Newtonian attraction may be exactly balanced by a centrifugal force. For example, a planar central configuration may rotate uniformly around its center of mass, thus defining a relative equilibrium of the n-body problem. Astronomical examples with n=3 are the Trojan asteroids and other bodies at some Lagrange point.
In 1995 I proved that there are only four types of central configurations of 4 bodies with equal masses, by using polynomial elimination. Here I wish to determine the central configurations of 5 bodies which have 3 bodies on a line. I will solve this problem in a symmetric case, where the existence result was published by Marian Gidea and Jaume Llibre in 2010, and independently by Kuo-Chang Chen and Jun-Shian Hsiao in 2011.

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Abstract. We propose a new method for retrieving the algebraic structure of a generic alternant code given an arbitrary generator matrix, provided certain conditions are met. We then discuss how this challenges the security of the McEliece cryptosystem instantiated with this family of codes. The central object of our work is the quadratic hull related to a linear code, defined as the intersection of all quadrics passing through the columns of a given generator or parity-check matrix, where the columns are considered as points in the affine or projective space. The geometric properties of this object reveal important information about the internal algebraic structure of the code. This is particularly evident in the case of generalized Reed-Solomon codes, whose quadratic hull is deeply linked to a well-known algebraic variety called the rational normal curve. By utilizing the concept of Weil restriction of affine varieties, we demonstrate that the quadratic hull of a generic dual alternant code inherits many interesting features from the rational normal curve, on account of the fact that alternant codes are subfield-subcodes of generalized Reed-Solomon codes. If the rate of the generic alternant code is sufficiently high, this allows us to construct a polynomial-time algorithm for retrieving the underlying generalized Reed-Solomon code from which the alternant code is defined, which leads to an efficient key-recovery attack against the McEliece cryptosystem when instantiated with this class of codes. Finally, we discuss the generalization of this approach to Algebraic-Geometry codes and Goppa codes.

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