Vol. 306, No. 2, 2020

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Explicit polynomial bounds on prime ideals in polynomial rings over fields

William Simmons and Henry Towsner

Vol. 306 (2020), No. 2, 721–754

Consider an ideal I k[x1,,xn] of a polynomial ring over a field with the property that for some b, if fg I for f,g of degree b, then f I or  g I. It is known that if b is sufficiently large, then I is prime. We construct an explicit bound on b, polynomial in the degree of the generators of I (the existence of such a bound was established by Schmidt-Göttsch in 1989). We also give a similar bound for detecting maximal ideals in  k[x1,,xn].

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uniform bounds, prime ideals, maximal ideals, proof mining, Gröbner bases
Mathematical Subject Classification 2010
Primary: 12Y05
Secondary: 12L10
Received: 7 June 2019
Revised: 28 January 2020
Accepted: 30 January 2020
Published: 13 July 2020
William Simmons
Department of Mathematics and Computer Science
Hobart and William Smith Colleges
Geneva, NY
United States
Henry Towsner
Department of Mathematics
University of Pennsylvania
Philadelphia, PA
United States