Bounds in the theory of polynomial rings over fields. A nonstandard approach. (English) Zbl 0539.13011

van den Dries, Lou; Schmidt, K.

Invent. Math. 76, 77-91 (1984).

In this paper nonstandard methods are used to show the existence of bounds for polynomial rings over fields of which the following are typical: Let \(f,f_ 1,...,f_ m\in K[X_ 1,...,X_ n]=K[X]\) of degree \(\leq d\), K a field. (a) \(f\in(f_ 1,...,f_ m)\) iff \(f=\Sigma f_ ih_ i\) for certain \(h_ i\in K[X]\) of degree \(\leq A\); (b) \((f_ 1,...,f_ m)\) is prime or contains 1 iff \[ gh\in(f_ 1,...,f_ m)\Rightarrow g\in(f_ 1,...,f_ m)\quad or\quad h\in(f_ 1,...,f_ m) \] holds for all g, \(h\in K[X]\) of degree \(\leq P\). - Here A and P are bounds depending only on (n,d) (not on K, m, or the polynomials in question).
The results are proved by considering the ring extension \(K[X]\subset K[X]_{int}\), K an internal field in some nonstandard structure, \(K[X]_{int}\) the ring of internal polynomials in \(X_ 1,...,X_ n\) over K. (a) follows from the faithful flatness of this extension, (b) from the results that prime ideals of K[X] remain prime when extended to \(K[X]_{int}\). - By concentrating on the existence of bounds (rather than their construction) many complications of the original constructive approach are avoided.

MSC:

13F20	Polynomial rings and ideals; rings of integer-valued polynomials
12E05	Polynomials in general fields (irreducibility, etc.)
03H99	Nonstandard models
13B25	Polynomials over commutative rings

Keywords:

bounds for degree of polynomials; polynomial rings over fields; internal field; nonstandard structure

References:

[1]	Bourbaki, N.: Commutative algebra. Paris: Hermann 1972 · Zbl 0279.13001
[2]	van den Dries, L.: Model theory of fields. Thesis, Utrecht 1978 · Zbl 0986.03034
[3]	van den Dries, L.: Bounds and algorithms in the theory of polynomial ideals. In: Logic colloquium 1978, (Boffa, M., Van Dalen, D., McAloon, K., eds.) North-Holland 1979
[4]	van den Dries, L., Wilkie, A.J.: Gromov’s theorem concerning groups of polynomial growth and elementary logic. To appear in J. of Algebra · Zbl 0552.20017
[5]	Hermann, G.: Die Frage der endlich vielen Schritte in der Theorie der Polynomideale. Math. Ann.95, 736-788 (1926) · JFM 52.0127.01
[6]	Hochster, M.: Some applications of the Frobenius in characteristic 0. Bull. AMS84, 886-912 (1978) · Zbl 0421.14001
[7]	Keisler, H.J.: Foundations of infinitesimal calculus. Boston: Prindle, Weber & Schmidt 1976 · Zbl 0333.26001
[8]	Lang, S.: Algebra. Reading, Mass.: Addison-Wesley 1965 · Zbl 0193.34701
[9]	Robinson, A.: Metamathematical problems. J. Symbolic Logic38, 500-516 (1973) · Zbl 0289.02002
[10]	Robinson, A.: On bounds in the theory of polynomial ideals. Selected questions of algebra and logic. (A collection dedicated to the memory of A.I. Mal’cev). In: Abraham Robinson, Selected Papers, vol. 1, Model Theory and Algebra, pp. 482-489, Yale Un. Press 1979
[11]	Roquette, P.: Nonstandard aspects of Hilbert’s Irreducibility Theorem. In: Model Theory and Algebra. A memorial Tribute to Abraham, Robinson, Saracino, D., Weispfenning, V. (eds.). Lecture Notes in Mathematics, vol. 498, pp. 231-275. Berlin-Heidelberg-New York: Springer 1975
[12]	Schmidt, K.: Modelltheoretische Methoden in der Algebraischen Geometrie. Diplomarbeit, Kiel 1979
[13]	Seidenberg, A.: Constructions in algebra. Trans. AMS197, 273-313 (1974) · Zbl 0356.13007
[14]	Seidenberg, A.: Construction of the integral closure of a finite integral domain. II. Proc. AMS52, 368-372 (1975) · Zbl 0333.13004
[15]	Seidenberg, A.: Constructions in a polynomial ring over the ring of integers. American J. of Math.100, 685-703 (1978) · Zbl 0416.13013
[16]	Stolzenberg, G.: Constructive Normalization of an Algebraic Variety. Bull. AMS74, 595-599 (1968) · Zbl 0164.04202