# Multiple point

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.

#### References

 [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: http://encyclopediaofmath.org/index.php?title=Multiple_point&oldid=34233
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article