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

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
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### 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) · doi:10.1007/BF01206635 [6] Hochster, M.: Some applications of the Frobenius in characteristic 0. Bull. AMS84, 886-912 (1978) · Zbl 0421.14001 · doi:10.1090/S0002-9904-1978-14531-5 [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 · doi:10.2307/2273049 [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 · doi:10.1090/S0002-9947-1974-0349648-2 [14] Seidenberg, A.: Construction of the integral closure of a finite integral domain. II. Proc. AMS52, 368-372 (1975) · Zbl 0333.13004 · doi:10.1090/S0002-9939-1975-0424783-5 [15] Seidenberg, A.: Constructions in a polynomial ring over the ring of integers. American J. of Math.100, 685-703 (1978) · Zbl 0416.13013 · doi:10.2307/2373905 [16] Stolzenberg, G.: Constructive Normalization of an Algebraic Variety. Bull. AMS74, 595-599 (1968) · Zbl 0164.04202 · doi:10.1090/S0002-9904-1968-12023-3
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