Victor Delage. Ens Rennes
From regulous to rational bounded functions
Sala de Seminarios, Dpto de Matemáticas. Las Palmeras 3425, Universidad de Chile
Abstract:
A rational function (quotient of polynomials from $k[X_1, ... X_n]$) is called regular when it lies in a regular ring. For an algebraically closed field k, a regular function has to be polynomial, but when studying the real case, some new functions appear: the ones whose denominators have no zeros, like $\frac{1}{1+X^2}$.
Here the regular functions extend quite naturally to regulous functions: when the denominator may have some zero, but when it happens, so does the numerator "in a stronger way" and the function is still continuous. Regulous functions have some very interesting properties, both algebraic and geometric, like nœtherianity of the topology, radical principality or Cartan's theorems A and B. An arising question is then whether bigger function rings may keep interesting properties; and we propose to loosen the continuity hypothesis to make it a bounded hypothesis; and see what happens.