Diophantine equations, Hilbert series, and undecidable spaces. (English) Zbl 0584.13012

For a long time it was an open question whether the Poincaré series of local rings and the Poincaré series of loop spaces were rational functions. These questions were answered in the negative by the author [Ann. Math., II. Ser. 115, 1-33 (1982; Zbl 0454.55004)]. In this paper the author continues to study how ”wild” these series can be. By a result of Roos it is enough to construct finitely presented Hopf algebras with ”bad” Hilbert series, these Hilbert series are reflected in the Poincaré series of corresponding local rings and Poincaré series of corresponding loop spaces. The paper under review develops a way to model Diophantine equations within the category of graded Hopf algebras. The ”solution series” of an arbitrary Diophantine equation is built in the Hilbert series for a corresponding graded Hopf algebra. Using results of Matiyasevic the author then can construct local rings and loop spaces with unexpectedly complex Poincaré series.
Reviewer: R.Fröberg


13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
55P62 Rational homotopy theory
13L05 Applications of logic to commutative algebra
55P35 Loop spaces
13H99 Local rings and semilocal rings
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