An analytic approach to the degree bound in the Nullstellensatz

Kwon, Hyun-Kyoung; Netyanun, Anupan; Trent, Tavan

doi:10.1090/proc/12781

An analytic approach to the degree bound in the Nullstellensatz
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by Hyun-Kyoung Kwon, Anupan Netyanun and Tavan T. Trent PDF

Proc. Amer. Math. Soc. 144 (2016), 1145-1152 Request permission

Abstract:

The Bezout version of Hilbert’s Nullstellensatz states that polynomials without a common zero form the unit ideal. In this paper, we start with a finite number of univariate polynomials and consider the polynomials that show up as a result of the Nullstellensatz. We present a simple analytic method of obtaining a bound for the degrees of these polynomials. Our result recovers W. D. Brownawell’s bound and is consistent with that of J. Kollár in the univariate case. The proof involves some well-known results on the analyticity of complex-valued functions.

References

Mats Andersson, Topics in complex analysis, Universitext, Springer-Verlag, New York, 1997. MR 1419088
W. Dale Brownawell, Bounds for the degrees in the Nullstellensatz, Ann. of Math. (2) 126 (1987), no. 3, 577–591. MR 916719, DOI 10.2307/1971361
Lennart Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. (2) 76 (1962), 547–559. MR 141789, DOI 10.2307/1970375
Grete Hermann, Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, Math. Ann. 95 (1926), no. 1, 736–788 (German). MR 1512302, DOI 10.1007/BF01206635
David Hilbert, Ueber die vollen Invariantensysteme, Math. Ann. 42 (1893), no. 3, 313–373 (German). MR 1510781, DOI 10.1007/BF01444162
Lars Hörmander, An introduction to complex analysis in several variables, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0203075
János Kollár, Sharp effective Nullstellensatz, J. Amer. Math. Soc. 1 (1988), no. 4, 963–975. MR 944576, DOI 10.1090/S0894-0347-1988-0944576-7
D. W. Masser and G. Wüstholz, Fields of large transcendence degree generated by values of elliptic functions, Invent. Math. 72 (1983), no. 3, 407–464. MR 704399, DOI 10.1007/BF01398396
R. Mortini and R. Rupp, Corona-type theorems and division in some function algebras on planar domains, The Corona Problem: Fields Institute Communications, Vol. 72 (2014), Springer, New York, 127-151.
Raghavan Narasimhan, Complex analysis in one variable, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 781130, DOI 10.1007/978-1-4757-1106-6
J. Ryle and T. T. Trent, A corona theorem for certain subalgebras of $H^\infty (\Bbb D)$ II, Houston J. Math. 38 (2012), no. 4, 1277–1295. MR 3019035, DOI 10.1111/j.1540-6261.1983.tb02295.x
Martín Sombra, A sparse effective Nullstellensatz, Adv. in Appl. Math. 22 (1999), no. 2, 271–295. MR 1659402, DOI 10.1006/aama.1998.0633
Mary Lawrence Thompson, TOPICS IN THE IDEAL THEORY OF COMMUTATIVE NOETHERIAN RINGS, ProQuest LLC, Ann Arbor, MI, 1986. Thesis (Ph.D.)–University of Michigan. MR 2634812
Tavan T. Trent, A new estimate for the vector valued corona problem, J. Funct. Anal. 189 (2002), no. 1, 267–282. MR 1887635, DOI 10.1006/jfan.2001.3842
Tavan Trent and Xinjun Zhang, A matricial corona theorem, Proc. Amer. Math. Soc. 134 (2006), no. 9, 2549–2558. MR 2213732, DOI 10.1090/S0002-9939-06-08172-X