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P. Gillibert:
"Julia Robinson's Numbers";
Vortrag: 19th Annual Seminar on Logic and its Applications held at the Jadavpur University of Calcutta, India, Jadavpur University, Calcutta, India (eingeladen); 13.10.2017 - 15.10.2017.

There are many examples of rings of algebraic integers with undecidable first order theory. Let K be an algebraic extension of Q. We denote by O(K) the ring of algebraic integers in K. Julia Robinson proved in 1959 that if K is of finite degree then the first order theory of the ring O(K) is undecidable, and similarly the first order theory of the field K is undecidable.

There is no such generic results for extensions of infinite degree. However Julia Robinson (1962) gave a powerful tool in order to prove the undecidability of the first order theory of various algebraic integer rings. An algebraic number x is totally real if all its conjugates are real numbers. Denote by R the field of all totally real algebraic numbers. A field K is totally real if K is included in R.

Assume that K is totally real. Denotes by J(K) the set of all real number r such that there are infinitely many x in O(K) such that all conjugates of x are in [0,r]. Note that J(K) is an interval. For example if K is of finite degree then J(K) is reduced to {infinity}. Xavier Vidaux and Carlos Videla (2016) observe that if K has the Northcott property then J(K) is {infinity}. Julia Robinson noticed that J(R)=[4,+infinity]. Indeed if z is a root of unity and z' its complex conjugatem then 2+z+z' is totally real and all its conjugates are in [0,4]. On the other hand, it follows from a result of Issai Schur (1918) or Leopold Kronecker (1857), that for each 0<r<4 there are only finitely many totally real algebraic integers with all conjugates in [0,r].

Julia Robinson (1962) proved that if J(K) is closed then the first order theory of O(K) is undecidable. In particular all previously mentioned ring have an undecidable first order theory. The Julia Robinson number of O(K) is the the infinum of J(K).

So far the only known examples of Julia Robinson's number are 4 and infinity. We give here new examples (in particular if t is a square-free odd integer then 8t is the Julia Robinson's number). Moreover in all the given examples the Julia Robinson's interval is closed, hence the first order theory is undecidable.