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Abel's Curve Theorem

Abel's curve theorem states that the sum of the values of an integral of the "first" or "second" sort

int_(x_0,y_0)^(x_1,y_1)(Pdx)/Q+...+int_(x_0,y_0)^(x_N,y_N)(Pdx)/Q=F(z)

and

(P(x_1,y_1))/(Q(x_1,y_1))(dx_1)/(dz)+...+(P(x_N,y_N))/(Q(x_N,y_N))(dx_N)/(dz)=(dF)/(dz),

from a fixed point to the points of intersection with an algebraic curve depending rationally upon any number of parameters is a rational function of those parameters.

This intersection formulation is related to Abel's addition theorem, which reduces an arbitrary sum of Abelian integrals to a number controlled by the curve genus, plus an elementary function (Kleiman 2004).

See also

Abel's Addition Theorem, Abelian Integral, Algebraic Curve

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References

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 277, 1959.Kleiman, S. L. "What Is Abel's Theorem Anyway?" In The Legacy of Niels Henrik Abel: The Abel Bicentennial, Oslo, 2002 (Ed. O. A. Laudal and R. Piene). Berlin, Germany: Springer, pp. 395-440, 2004. https://doi.org/10.1007/978-3-642-18908-1_11.

Referenced on Wolfram|Alpha

Abel's Curve Theorem

Cite this as:

Weisstein, Eric W. "Abel's Curve Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AbelsCurveTheorem.html

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