Root

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

(Subscript[(z;(∑_(j=0)^n)(Subscript[a, j])(z^j)), k]^(-1)) is the kth root of the polynomial equation (∑_(j=0)^n)(Subscript[a, j]) (z^j)==0.

Specific values

Specialized values

Values at fixed points

General characteristics

Domain and analyticity

(Subscript[(z;(∑_(j=0)^n)(Subscript[a, j])(z^j)), k]^(-1)) is an analytic function of the (Subscript[a, j])  (0≤j≤n) in (ℂ^(n+1)).

Symmetries and periodicities

Symmetry

No symmetry

Periodicity

No periodicity

Poles and essential singularities

In most cases there are no poles, but poles up to order n-1can be present.

Branch points

For generic values (Subscript[a, j]), it has n-1 branchpoints of order 2. At most (Subscript[(z;(∑_(j=0)^n)(Subscript[a, j])(z^j)), k]^(-1)) can have one branch point of order n.

Branch cuts

The location of the branch cuts is complicated. Generically the branch cuts run asymptotically radially outwards and do not connect branch points.

Limit representations

Differentiation

Low-order differentiation

With respect to (Subscript[a, k])

Representations through equivalent functions

With related functions

Inequalities

Theorems

Salem number

The smallest known Salem number (a real algebraic integer greater than 1 whose conjugates have absolute value  ≤1, at least one conjugate having absolute value ==1) is given by (Subscript[(z;1+z-(z^3)-(z^4)-(z^5)-(z^6)-(z^7)+(z^9)+(z^10)), 2]^(-1)) ∝1.17628….

Pisot numbers

Raising simple algebraic numbers to a high power can yield numbers very near to integers. Example:   ((((Subscript[(z;-1-z+(z^3)), 1]^(-1))))^10000) is within (10^(-611)) of an integer.

Unequality for rational numbers

For any algebraic α of degree greater than 1 there exists a c(α) such that |(p/q)-α|>((c(α))/(q^2)) for all rational numbers (p/q).

The Lagrange points of the restricted three-body problem

The Lagrange points (Subscript[L, 1]), (Subscript[L, 2]), and (Subscript[L, 3])of the restricted three-body problem with potential V(x,y)==-(1/2) ((x^2)+(y^2))-((1-μ)/(Sqrt[(y^2)+(((x-(Subscript[x, 1])))^2)]))-(μ/(Sqrt[(y^2)+(((x-(Subscript[x, 2])))^2)])) are given by {(OverTilde[(Subscript[x, i])]),0}, where the (OverTilde[(Subscript[x, i])]) are the real solutions of the quintic polynomial equation

(x^5)-2  ((Subscript[x, 1])+(Subscript[x, 2]))(x^4)+((Subscript[x, 1]^2)+4 (Subscript[x, 2]) (Subscript[x, 1])+(Subscript[x, 2]^2)) (x^3)-(2 (Subscript[x, 1]) (Subscript[x, 2]) ((Subscript[x, 1])+(Subscript[x, 2]))+1) (x^2)+((Subscript[x, 1]^2) (Subscript[x, 2]^2)+2 μ (Subscript[x, 1])-2 (μ-1) (Subscript[x, 2])) x+(μ-1) (Subscript[x, 2]^2)-μ (Subscript[x, 1]^2)==0.

Gauss-Lucas theorem

max(|(Subscript[(z;(∑_(j=0)^n)(Subscript[a, j]) (z^j)), 1]^(-1))|,|(Subscript[(z;(∑_(j=0)^n)(Subscript[a, j]) (z^j)), 2]^(-1))|,…, |(Subscript[(z;(∑_(j=0)^n)(Subscript[a, j]) (z^j)), n]^(-1))|)≥max(|(Subscript[(z;(∂/(∂z))(∑_(j=0)^n)(Subscript[a, j]) (z^j)), 1]^(-1))|,|(Subscript[(z;(∂/(∂z))(∑_(j=0)^n)(Subscript[a, j]) (z^j)), 2]^(-1))|,…, |(Subscript[(z;(∂/(∂z))(∑_(j=0)^(n-1))(Subscript[a, j]) (z^j)), n-1]^(-1))|).

The hard hexagon entropy constant

The hard hexagon entropy constant is an algebraic number of degree 24.

History

– J. L. Lagrange (1767-1772)
– P. Ruffini (1799)
– C. F. Gauss (1801)
– N. H. Abel (1826)
– J. C. F. Sturm (1829)
– E. Galois (1832)
– G. Eisenstein (1844)
– J. Cockle,  J. K. Thomae (1869)
– C. Jordan (1870)
– F. von Lindermann  (1884,1892)
– L. Kronecker (1890,1891)
– R. H. Mellin (1915)
– R. Birkeland (1905-1925)

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