Interpolation over \(\mathbb{Z}\) and torsion in class groups. (English) Zbl 1502.13003

Berman, John D.; Erman, Daniel

J. Commut. Algebra 14, No. 3, 309-314 (2022).

Summary: We prove an interpolation result for homogeneous polynomials over the integers, or more generally for PIDs with finite residue fields. Previous proofs of this result use the well-known but nontrivial fact that class groups of rings of integers are torsion. We provide an independent proof using elementary techniques.

MSC:

13A02	Graded rings
11R29	Class numbers, class groups, discriminants

Keywords:

class groups; interpolation; polynomials

Software:

Macaulay2

References:

[1]	J. Bruce and D. Erman, “A probabilistic approach to systems of parameters and Noether normalization”, Algebra Number Theory 13:9 (2019), 2081-2102. · Zbl 1431.13011 · doi:10.2140/ant.2019.13.2081
[2]	T. Chinburg, L. Moret-Bailly, G. Pappas, and M. J. Taylor, “Finite morphisms to projective space and capacity theory”, J. Reine Angew. Math. 727 (2017), 69-84. · Zbl 1454.14020 · doi:10.1515/crelle-2014-0089
[3]	D. Eisenbud, Commutative algebra: with a view toward algebraic geometry, Graduate Texts in Mathematics 150, Springer, 1995. · Zbl 0819.13001 · doi:10.1007/978-1-4612-5350-1
[4]	D. Eisenbud, The geometry of syzygies: a second course in commutative algebra and algebraic geometry, Graduate Texts in Mathematics 229, Springer, 2005. · Zbl 1066.14001
[5]	J. Fresnel and M. Matignon, “Good rings and homogeneous polynomials”, 2019. · Zbl 1510.13002
[6]	O. Gabber, Q. Liu, and D. Lorenzini, “Hypersurfaces in projective schemes and a moving lemma”, Duke Math. J. 164:7 (2015), 1187-1270. · Zbl 1432.14002 · doi:10.1215/00127094-2877293
[7]	O. Goldman, “On a special class of Dedekind domains”, Topology 3:1 (1964), 113-118. · Zbl 0135.08005 · doi:10.1016/0040-9383(64)90009-6
[8]	D. R. Grayson and M. E. Stillman, “Macaulay2, a software system for research in algebraic geometry”, available at http://www.math.uiuc.edu/Macaulay2.
[9]	International Mathematical Olympiad, “Shortlisted Problems, IMO 2017”, available at https://www.imo-official.org/problems/IMO2017SL.pdf.
[10]	D. Khurana, T. Y. Lam, and Z. Wang, “Rings of square stable range one”, J. Algebra 338 (2011), 122-143 · Zbl 1237.19004 · doi:10.1016/j.jalgebra.2011.03.034