Polynomial lemniscate

Last updated November 09, 2023

|
z
6
+
z
5
+
z
4
+
z
3
+
{\displaystyle |z^{6}+z^{5}+z^{4}+z^{3}+}
z
2
+
z
+
1
|
=
1
{\displaystyle z^{2}+z+1|=1} Cyc7.png — $|z^{6}+z^{5}+z^{4}+z^{3}+$ $z^{2}+z+1|=1$

In mathematics, a polynomial lemniscate or polynomial level curve is a plane algebraic curve of degree 2n, constructed from a polynomial p with complex coefficients of degree n.

For any such polynomial p and positive real number c, we may define a set of complex numbers by $|p(z)|=c.$ This set of numbers may be equated to points in the real Cartesian plane, leading to an algebraic curve ƒ(x, y) = c² of degree 2n, which results from expanding out $p(z){\bar {p}}({\bar {z}})$ in terms of z = x + iy.

When p is a polynomial of degree 1 then the resulting curve is simply a circle whose center is the zero of p. When p is a polynomial of degree 2 then the curve is a Cassini oval.

Erdős lemniscate

Erdos lemniscate of degree ten and genus six Erdos5.png — Erdős lemniscate of degree ten and genus six

A conjecture of Erdős which has attracted considerable interest concerns the maximum length of a polynomial lemniscate ƒ(x, y) = 1 of degree 2n when p is monic, which Erdős conjectured was attained when p(z) = zⁿ − 1. This is still not proved but Fryntov and Nazarov proved that p gives a local maximum.^[1] In the case when n = 2, the Erdős lemniscate is the Lemniscate of Bernoulli

(x^{2}+y^{2})^{2}=2(x^{2}-y^{2})\,

and it has been proven that this is indeed the maximal length in degree four. The Erdős lemniscate has three ordinary n-fold points, one of which is at the origin, and a genus of (n − 1)(n − 2)/2. By inverting the Erdős lemniscate in the unit circle, one obtains a nonsingular curve of degree n.

Generic polynomial lemniscate

In general, a polynomial lemniscate will not touch at the origin, and will have only two ordinary n-fold singularities, and hence a genus of (n − 1)². As a real curve, it can have a number of disconnected components. Hence, it will not look like a lemniscate, making the name something of a misnomer.

An interesting example of such polynomial lemniscates are the Mandelbrot curves. If we set p₀ = z, and p_n = p_n−1² + z, then the corresponding polynomial lemniscates M_n defined by |p_n(z)| = 2 converge to the boundary of the Mandelbrot set.^[2] The Mandelbrot curves are of degree 2ⁿ⁺¹.^[3]

Notes

↑ Fryntov, A; Nazarov, F (2008). "New estimates for the length of the Erdos-Herzog-Piranian lemniscate". Linear and Complex Analysis. 226: 49–60. arXiv: 0808.0717 . Bibcode:2008arXiv0808.0717F.
↑ Desmos.com - The Mandelbrot Curves
↑ Ivancevic, Vladimir G.; Ivancevic, Tijana T. (2007), High-Dimensional Chaotic and Attractor Systems: A Comprehensive Introduction, Springer, p. 492, ISBN 9781402054563 .

Related Research Articles

Algebraic geometry is a branch of mathematics which classically studies zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros.

In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign =. The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, while in English, any well-formed formula consisting of two expressions related with an equals sign is an equation.

The Mandelbrot set is a two-dimensional set with a relatively simple definition that exhibits great complexity, especially as it is magnified. It is popular for its aesthetic appeal and fractal structures. The set is defined in the complex plane as the complex numbers $for which the function does not diverge to infinity when iterated starting at, i.e., for which the sequence,, etc., remains bounded in absolute value.$

In mathematics, a transcendental number is a real or complex number that is not algebraic – that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are $π$ and $e$ .

In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the original polynomial. The discriminant is widely used in polynomial factoring, number theory, and algebraic geometry.

Bézout's theorem is a statement in algebraic geometry concerning the number of common zeros of $n$ polynomials in $n$ indeterminates. In its original form the theorem states that in general the number of common zeros equals the product of the degrees of the polynomials. It is named after Étienne Bézout.

In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties.

In mathematics, the Weil conjectures were highly influential proposals by André Weil (1949). They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory.

Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition.

In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points $F 1$ and $F 2$ , known as foci, at distance $2 c$ from each other as the locus of points $P$ so that $PF 1 \cdot PF 2 = c 2$ . The curve has a shape similar to the numeral 8 and to the ∞ symbol. Its name is from lemniscatus, which is Latin for "decorated with hanging ribbons". It is a special case of the Cassini oval and is a rational algebraic curve of degree 4.

In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation $h (x, y, t) = 0$ can be restricted to the affine algebraic plane curve of equation $h (x, y, 1) = 0$ . These two operations are each inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered.

In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities and allowing "varieties" defined over any commutative ring.

In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures. Étale cohomology theory can be used to construct ℓ-adic cohomology, which is an example of a Weil cohomology theory in algebraic geometry. This has many applications, such as the proof of the Weil conjectures and the construction of representations of finite groups of Lie type.

In mathematics, an algebraic equation or polynomial equation is an equation of the form $, where P is a polynomial with coefficients in some field, often the field of the rational numbers. For example, is an algebraic equation with integer coefficients and$

In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field of real numbers, a rational point is more commonly called a real point.

In algebraic geometry, a lemniscate is any of several figure-eight or $\infty$ -shaped curves. The word comes from the Latin lēmniscātus meaning "decorated with ribbons", from the Greek λημνίσκος meaning "ribbon", or which alternatively may refer to the wool from which the ribbons were made.

A complex quadratic polynomial is a quadratic polynomial whose coefficients and variable are complex numbers.

Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Part of the inspiration comes from complex dynamics, the study of the iteration of self-maps of the complex plane or other complex algebraic varieties. Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, $p$ -adic, or algebraic points under repeated application of a polynomial or rational function. A fundamental goal is to describe arithmetic properties in terms of underlying geometric structures.

In mathematics, the polynomial method is an algebraic approach to combinatorics problems that involves capturing some combinatorial structure using polynomials and proceeding to argue about their algebraic properties. Recently, the polynomial method has led to the development of remarkably simple solutions to several long-standing open problems. The polynomial method encompasses a wide range of specific techniques for using polynomials and ideas from areas such as algebraic geometry to solve combinatorics problems. While a few techniques that follow the framework of the polynomial method, such as Alon's Combinatorial Nullstellensatz, have been known since the 1990s, it was not until around 2010 that a broader framework for the polynomial method has been developed.

References

Alexandre Eremenko and Walter Hayman, On the length of lemniscates, Michigan Math. J., (1999), 46, no. 2, 409–415
O. S. Kusnetzova and V. G. Tkachev, Length functions of lemniscates, Manuscripta Math., (2003), 112, 519–538

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Fryntov, A; Nazarov, F (2008). "New estimates for the length of the Erdos-Herzog-Piranian lemniscate". Linear and Complex Analysis. 226: 49–60. arXiv: 0808.0717 . Bibcode:2008arXiv0808.0717F.

[2] Desmos.com - The Mandelbrot Curves

[3] Ivancevic, Vladimir G.; Ivancevic, Tijana T. (2007), High-Dimensional Chaotic and Attractor Systems: A Comprehensive Introduction, Springer, p. 492, ISBN 9781402054563 .

[1]

[2]

[3]