Multiple point

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of a planar curve $F(x,y)=0$

A singular point at which the partial derivatives of order up to and including $n$ vanish, but where at least one partial derivative of order $n+1$ does not vanish. For example, if $F(x_0,y_0)=0$, $F_x'(x_0,y_0)=0$, $F_{yy}''(x_0,y_0)$ does not vanish, the multiple point $M(x_0,y_0)$ is called a double point; if the first and second partial derivatives vanish at $M(x_0,y_0)$, but at least one third derivative does not, the multiple point is called a triple point; etc.


[a1] J.L. Coolidge, "Algebraic plane curves" , Dover, reprint (1959) MR0120551 Zbl 0085.36403
[a2] D. Hilbert, S.E. Cohn-Vossen, "Geometry and the imagination" , Chelsea (1952) pp. 173 (Translated from German) MR0046650 Zbl 0047.38806
[a3] P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) MR0507725 Zbl 0408.14001
[a4] W. Fulton, "Algebraic curves" , Benjamin (1969) pp. 66 MR0313252 MR0260752 Zbl 0194.21901 Zbl 0181.23901
How to Cite This Entry:
Multiple point. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article