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Acnode

An acnode is an isolated point in the solution set of a polynomial equation in two real variables. Equivalent terms are isolated point and hermit point.^[1]

For example the equation

f(x,y)=y^{2}+x^{2}-x^{3}=0

has an acnode at the origin, because it is equivalent to

y^{2}=x^{2}(x-1)

and $x^{2}(x-1)$ is non-negative only when $x$ ≥ 1 or $x=0$ . Thus, over the real numbers the equation has no solutions for $x<1$ except for (0, 0).

In contrast, over the complex numbers the origin is not isolated since square roots of negative real numbers exist. In fact, the complex solution set of a polynomial equation in two complex variables can never have an isolated point.

An acnode is a critical point, or singularity, of the defining polynomial function, in the sense that both partial derivatives $\partial f \over \partial x$ and $\partial f \over \partial y$ vanish. Further the Hessian matrix of second derivatives will be positive definite or negative definite, since the function must have a local minimum or a local maximum at the singularity.

References

^ Hazewinkel, M. (2001) [1994], "Acnode", Encyclopedia of Mathematics, EMS Press

Porteous, Ian (1994). Geometric Differentiation. Cambridge University Press. ISBN 978-0-521-39063-7.

Topics in algebraic curves

Rational curves

Elliptic curves

Analytic theory	Elliptic function Elliptic integral Fundamental pair of periods Modular form
Arithmetic theory	Counting points on elliptic curves Division polynomials Hasse's theorem on elliptic curves Mazur's torsion theorem Modular elliptic curve Modularity theorem Mordell–Weil theorem Nagell–Lutz theorem Supersingular elliptic curve Schoof's algorithm Schoof–Elkies–Atkin algorithm
Applications	Elliptic curve cryptography Elliptic curve primality

Higher genus

Plane curves

Riemann surfaces

Constructions

Structure of curves

Divisors on curves	Abel–Jacobi map Brill–Noether theory Clifford's theorem on special divisors Gonality of an algebraic curve Jacobian variety Riemann–Roch theorem Weierstrass point Weil reciprocity law
Moduli	ELSV formula Gromov–Witten invariant Hodge bundle Moduli of algebraic curves Stable curve
Morphisms	Hasse–Witt matrix Riemann–Hurwitz formula Prym variety Weber's theorem (Algebraic curves)
Singularities	Acnode Crunode Cusp Delta invariant Tacnode
Vector bundles	Birkhoff–Grothendieck theorem Stable vector bundle Vector bundles on algebraic curves

This algebraic geometry–related article is a stub. You can help Wikipedia by expanding it.

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