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

An acnode at the origin (curve described in text)

For example the equation

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

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.

Singular point of a curve
Crunode
Cusp
Tacnode

References

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

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Topics in algebraic curves
Rational curves

Five points determine a conic Projective line Rational normal curve Riemann sphere Twisted cubic

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

De Franchis theorem Faltings's theorem Hurwitz's automorphisms theorem Hurwitz surface Hyperelliptic curve

Plane curves

AF+BG theorem Bézout's theorem Bitangent Cayley–Bacharach theorem Conic section Cramer's paradox Cubic plane curve Fermat curve Genus–degree formula Hilbert's sixteenth problem Nagata's conjecture on curves Plücker formula Quartic plane curve Real plane curve

Riemann surfaces

Belyi's theorem Bring's curve Bolza surface Compact Riemann surface Dessin d'enfant Differential of the first kind Klein quartic Riemann's existence theorem Riemann–Roch theorem Teichmüller space Torelli theorem

Constructions

Dual curve Polar curve Smooth completion

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

Singularities

Acnode Crunode Cusp Delta invariant Tacnode

Vector bundles

Birkhoff–Grothendieck theorem Stable vector bundle Vector bundles on algebraic curves