A note on determinacy

Bruce, J. W.; Ruas, M. A. S.; Saia, M. J.

doi:10.1090/S0002-9939-1992-1101980-7

A note on determinacy
HTML articles powered by AMS MathViewer

by J. W. Bruce, M. A. S. Ruas and M. J. Saia

Proc. Amer. Math. Soc. 115 (1992), 865-871

DOI: https://doi.org/10.1090/S0002-9939-1992-1101980-7

PDF | Request permission

Abstract:

In this paper, we present a particularly simple and direct proof that the set of noncontact-sufficient ($\mathcal {K}$-sufficient) germs are of infinite codimension. Our proof gives, for each $k$, an integer $r$ with the property that almost all $r$-jets over any $k$-jet $z$ is $\mathcal {K}$-sufficient. Similar results are obtained for $\mathcal {A}$ or right-left equivalence when the source and target dimensions $(n,p)$ are (2,2) and (2,3).

References

J. Bochnak, Relèvement des jets, Séminaire Pierre Lelong (Analyse) (année 1970–1971), Lecture Notes in Math., Vol. 275, Springer, Berlin, 1972, pp. 106–118 (French). MR 0377971
J. W. Bruce, A. A. du Plessis, and C. T. C. Wall, Determinacy and unipotency, Invent. Math. 88 (1987), no. 3, 521–554. MR 884799, DOI 10.1007/BF01391830
James Damon, Finite determinacy and topological triviality. I, Invent. Math. 62 (1980/81), no. 2, 299–324. MR 595590, DOI 10.1007/BF01389162

Weighted homogeneous maps from the plane to the plane

G.-M. Greuel, Der Gauss-Manin-Zusammenhang isolierter Singularitäten von vollständigen Durchschnitten, Math. Ann. 214 (1975), 235–266. MR 396554, DOI 10.1007/BF01352108
John N. Mather, Generic projections, Ann. of Math. (2) 98 (1973), 226–245. MR 362393, DOI 10.2307/1970783

Quasi-homogeneous maps from

to

Andrew du Plessis, Genericity and smooth finite determinacy, Singularities, Part 1 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 295–312. MR 713068, DOI 10.1090/pspum/040.1/713068
C. T. C. Wall, Finite determinacy of smooth map-germs, Bull. London Math. Soc. 13 (1981), no. 6, 481–539. MR 634595, DOI 10.1112/blms/13.6.481
Klaus Wirthmüller, A note on contact sufficiency of jets, J. London Math. Soc. (2) 25 (1982), no. 1, 185–192. MR 645875, DOI 10.1112/jlms/s2-25.1.185