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Jonquière's Theorem

For an algebraic curve, the total number of groups of a g_N^r consisting in a point of multiplicity k_1, one of multiplicity k_2, ..., one of multiplicity k_rho, where

sumk_i=N
(1)
sum(k_i-1)=r,
(2)

and where alpha_1 points have one multiplicity, alpha_2 another, etc., and

Pi=k_1k_2...k_rho
(3)

is

(Pip(p-1)...(p-rho))/(alpha_1!alpha_2!...)[Pi/(p-rho)-(sum_(i)(partialPi)/(partialk_i))/(p-rho+1)+(sum_(ij)(partial^2Pi)/(partialk_ipartialk_j))/(p-rho+2)+...].
(4)

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References

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 288, 1959.Jonquière, A. "Mémoire sue les contacts multiples d'ordre quelconque de courbes, etc." J. reine angew. Math. 66, 288, 1866.Zeuthen. Lehrbuch der abzählenden Methoden in der Geometrie. Leipzig, Germany: p. 240, 1914.

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Jonquière's Theorem

Cite this as:

Weisstein, Eric W. "Jonquière's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/JonquieresTheorem.html

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