Diophantine unsolvability for function fields over certain infinite fields of characteristic \(p\)
Since the negative answer to Hilbert's Tenth Problem was given in 1971, attention was given to various similar questions. The paper answers the following Question: Let \(\mathbb{F}\) be a field properly contained in the algebraic closure of a finite field \(\mathbb{F}_ p\) where \(p\) is an odd prime. Is there an algorithm to decide which polynomial equations \(f(x_ 1,\dots,x_ r)=0\) with coefficients in the function field \(\mathbb{F}_ p(t)\) do have solutions in \(\mathbb{F}(t)\) and which do not? The negative answer the paper gives to the question is typical for this type of problem: for all fields \(\mathbb{F}\) for which the answer is known (notably for \(\mathbb{F}\) = the field of reals, a finite field or \(\mathbb{C}(t)\) (\(\mathbb{C}\) the field of complex numbers)), it is negative. The problem remains open for all algebraically closed fields \(\mathbb{F}\). What the paper does is giving an existential definition of the relation ``the function \(x\) in \(\mathbb{F}(t)\) has the value 0 at \(t=0\)'', an existential definition here is one that consists of a number of existential quantifiers followed by an algebraic equation. Once such a definition is found, the proof of the result is known by previous results of the reviewer (an analogous situation holds in zero characteristic based on works by J. Denef). To construct the definition the authors use norm forms in conjunction with the fact that \(\mathbb{F}\) is not algebraically closed. The latter is crucial here, because, if \(\mathbb{F}\) is algebraically closed, any norm form represents all elements of \(\mathbb{F}(t)\) and is therefore useless in the context of the question.
scientific article; zbMATH DE number 130500