Separating the solution sets of analytical and polynomial systems  Miguel A. Goberna y Departamento de Estadstica e Investigacin Operativa, Universidad de Alicante, 03080 Alicante, Spain, email: mgoberna@ua.es Lidia HernÆndez z Facultad de Ciencias Fsico MatemÆticas, BenemØrita Universidad Autnoma de Puebla, 72500 Puebla, Mexico, email: lhernan@fcfm.buap.mx Maxim I. Todorov x Departamento de Fsica y MatemÆticas, Escuela de Ciencias, Universidad de las Americas , Sta. Catarina MÆrtir, 72820 Cholula, Puebla, Mexico, email: mtodorov@mail.udlap.mx Abstract. A linear inequality system with innitely many constraints is polynomial (analytical) if its index set is a compact interval of the real line and all its coe¢ cients are polynomial (analytical, respectively) functions of the index on this interval. This paper provides an example of analytical system whose solution set cannot be the solution set of any polynomial system. Key words: Linear semi-innite programming, linear systems, con- vex sets. AMS subject classication: 90C34, 15A39, 52A20. 1. Introduction A linear system  = fa 0 t x  b t ;t 2 T g is called analytical (polynomial ) if T is a compact interval in R and all the coe¢ cients b t = b(t) 2 R and a t = a(t)=(a 1 (t); :::; a n (t)) 0 2 R n are analytical (polynomial, respec- tively) functions of the index t. Analytical (polynomial) systems where introduced by Anderson and Lewis (1989) and by Goberna, HernÆndez and Todorov (2005), respectively. The main application eld of analytical and polynomial systems is linear semi-innite programming, the branch optimization which deals with the minimization of linear functionals under an arbitrary number of linear constraints. One the many applications gathered in Goberna  Short title: analytical vs. polynomial systems. y Research supported by DGES of Spain and FEDER of UE, Grant BFM2002- 04114-C02-01. z Research supported by CONACyT of Mexico, Grant 130036. x Research partially supported by CONACyT of Mexico, Grant 44003. c  2005 Kluwer Academic Publishers. Printed in the Netherlands. swp0001.tex; 9/03/2005; 10:36; p.1