Generic Finiteness for a Class of Symmetric Planar Central Configurations of the Six-Body Problem and the Six-Vortex Pro
- PDF / 475,611 Bytes
- 24 Pages / 439.37 x 666.142 pts Page_size
- 50 Downloads / 185 Views
Generic Finiteness for a Class of Symmetric Planar Central Configurations of the Six-Body Problem and the Six-Vortex Problem Thiago Dias1,2
· Bo-Yu Pan3
Received: 18 January 2019 / Revised: 22 May 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract A symmetric planar central configuration of the Newtonian six-body problem x is called cross central configuration if there are precisely four bodies on a symmetry line of x. We use complex algebraic geometry and Groebner basis theory to prove that for a generic choice of positive real masses m 1 , m 2 , m 3 , m 4 , m 5 = m 6 there is a finite number of cross central configurations. We also show one explicit example of a configuration in this class. A part of our approach is based on relaxing the output of the Groebner basis computations. This procedure allows us to obtain upper bounds for the dimension of an algebraic variety. We get the same results considering cross central configurations of the six-vortex problem. Keywords n-Body problem · n-Vortex problem · Central configuration · Groebner basis · Jacobian criterion · Elimination theory
The authors wish to thank Kuo-Chan Cheng for their hospitality and many stimulating conversations, to Alain Albouy for their continuous incentive, to Eduardo Leandro and Ya-Lun Tsai for their helpful remarks, and to the Department of Mathematics at the National Tsing Hua University and the Department of Mathematics at the Universidade Federal Rural de Pernambuco for their support. The first author was partly supported by the Ministry of Science and Technology of the Republic of China under the Grant MOST 107-2811-M-007-004.
B
Thiago Dias [email protected] Bo-Yu Pan [email protected]
1
Department of Mathematics, Universidade Federal Rural de Pernambuco, av. Don Manoel de Medeiros s/n, Dois Irmãos, Recife, PE 52171-900, Brasil
2
Present Address: Department of Mathematics, National Tsing Hua University, Hsinchu City 30013, Taiwan
3
Department of Mathematics, National Tsing Hua University, Hsinchu City 30013, Taiwan
123
Journal of Dynamics and Differential Equations
1 Introduction One of the leading open questions in the central configurations theory is the finiteness problem: for every choice of n point mass m 1 , . . . , m n , is the number of central configurations finite? Chazy and Wintner contributed significantly to the interest in this problem that appears in the Smale’s list for the Mathematicians of the twenty-first century [27]. Hampton and Moeckel used BKK theory to obtain the finiteness for central configurations of the four-body problem in the Newtonian case [13] and the vortex case [14]. Albouy and Kaloshin proved that for a choice of masses m 1 , . . . , m 5 in the complement of a codimension-2 algebraic variety on the mass space, there is a finite number of planar central configurations of the Newtonian fivebody problem [3]. They studied the behavior of unbounded singular sequences of normalized central configurations going to the infinity. In the last years,
Data Loading...
