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Artin Approximation and Solutions to Recursion Equations

Guest lecturer Prof. Herwig Hauser (Vienna)


Let f(x,y)=0 be a system of polynomial equations in two sets of variables x and y. The implicit function theorem gives a criterion to ensure the existence of a power series solution y(x) in terms of x nearby a given point a, say f(x,y(x))=0. Artin approximation is concerned with the existence of solutions in general and their degree of regularity (formal, convergent, algebraic power series). One version says that if we can find a solution y(x) up to sufficiently high degree, then we can find also a formal exact solution (which will even be convergent). All this can be expressed through recursion equations, where the coefficients c_k of the series y(x) are the unknowns and f(x,y(x))=0 prescribes the recursion. The result then says that once we know to solve for the c_k up to sufficiently large k, we can find all c_k, and these satisfy moreover certain growth conditions. In the lectures, we give a complete proof of Artin's result, based on the Weierstrass division theorem and on techniques from commutative algebra. We also discuss possible extensions and counter-examples to stronger formulations.


326.077, 1 hour, 1.5 ECTS, blocked lecture, March to June 2013


First block:

Thu, 21 March 2013, 11:00 – 13:00, MT 128,

       and 16:00 – 18:00, T 642;

Fri, 22 March 2013, 10:00 – 12:00, MT 130.


We will start on Thursday morning our lecture with a detailed discussion of what Artin approximation is all about, seen from different viewpoints: commutative algebra, infinite dimensional algebraic geometry, combinatorics, symbolic computation.  This will include various different versions of the main theorem (formal, approximate, convergent, algebraic, and nested power series solutions).

In the afternoon, we will make this more concrete by computing together several examples which will exhibit the main problems for proving the results while giving already some hints of how to achieve this.

In the Friday lecture, we will start with explaining the main tools for the proof of Artin approximation. It is based on an intricate induction on the number of variables, the induction step being performed by a tricky application of the Weierstrass division theorem.


Second block:

Thu, 11 April 2013, 11:00 - 13:00, MT 130,

       and 15:30 - 17:00, S3 058;

Fri, 12 April 2013, 10:00 - 12:00, S2 054.


We will recall the statement of the theorem, discuss various extensions of it, provide counter-examples to stronger statements, and start with developing the necessary techniques for the proof. We will also indicate the connection to combinatorics and to (infinite dimensional) differential or algebraic geometry.


Third block:

Thu, 25 April 2013, 11:00 - 13:00, S2 054,

       and 15:30 - 17:00, S2 054;

Fri, 26 April 2013, 10:00 - 12:00, S2 054.


We will discuss and prove the Weierstrass division theorem (and its generalization to ideals by Grauert and Hironaka), which will allow us to prove the linear version of the Approximation Theorem (flatness of K[[x]] over K{x}).

This will guide us already towards the proof of the general statement, for which we hope to sketch at least the main ideas.


Fourth block:

Thu, 16 May 2013, 11:00 - 13:00, MT 128,

       and 15:30 - 17:00, S2 054;

Fri, 17 May 2013, 10:00 - 12:00, S3 048.


We will develop the main arguments for the proof in the non-linear case.