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A note on the AF + BΦ theorem and the theory of local rings

Published online by Cambridge University Press:  24 October 2008

D. G. Northcott
D. G. Northcott
Affiliation:
The UniversitySheffield
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Extract

The so-called AF + BΦ theorem of Max Noether refers to a whole collection of results. In most of these, the object is to determine conditions under which a form H(x, y, z) will belong to the ideal generated by two given forms F(x, y, z) and Φ(x, y, z). This problem is of obvious importance in the theory of plane curves, and therefore it is customary to state the conditions in geometric language. To give one example, a particularly important form of the theorem may be stated as follows.*

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 51 , Issue 4 , October 1955 , pp. 545 - 550
Copyright
Copyright © Cambridge Philosophical Society 1955

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References

REFERENCES

(1)Northcott, D. G.Hilbert's function in a local ring. Quart. J. Math. (2), 4 (1953), 67–80.CrossRefGoogle Scholar
(2)Northcott, D. G.On the notion of a form ideal. Quart. J. Math. (2), 4 (1953), 221–9.CrossRefGoogle Scholar
(3)Northcott, D. G.The neighbourhoods of a local ring. J. Lond. math. Soc. 30 (1955), 360–75.CrossRefGoogle Scholar
(4)Samuel, P. La notion de multiplicité en algèbre et en géométrie (Thesis, Paris, 1951).Google Scholar
(5)Walker, R. J.Algebraic Curves (Princeton Mathematical Series, no. 13).Google Scholar