Principal equation form
In mathematics and, more specifically, in theory of equations, the principal form of an irreducible polynomial of degree at least three is a polynomial of the same degree n without terms of degrees n−1 and n−2, such that each roots of either polynomial is a rational function of a root of the other polynomial.
The principal form of a polynomial can be found by applying a suitable Tschirnhaus transformation to the given polynomial.
Definition
Let
be an irreducible polynomial of degree at least three.
Its principal form is a polynomial
together with a Tschirnhaus transformation of degree two
such that, if r is a root of f, is a root of .[1][2]
Expressing that does not has terms in and leads to a system of two equations in and , one of degree one and one of degree two. In general, this system has two solutions, giving two principal forms involving a square root. One passes from one principal form to the secong by changing the sign of the square root.[3][4]
Literature
- "Polynomial Transformations of Tschirnhaus", Bring and Jerrard, ACM Sigsam Bulletin, Vol 37, No. 3, September 2003
- F. Brioschi, Sulla risoluzione delle equazioni del quinto grado: Hermite — Sur la résolution de l'Équation du cinquiéme degré Comptes rendus —. N. 11. Mars. 1858. 1. Dezember 1858, doi:10.1007/bf03197334
- Bruce and King, Beyond the Quartic Equation, Birkhäuser, 1996.
References
- ^ Weisstein, Eric W. "Principal Quintic Form". mathworld.wolfram.com.
- ^ "The solution to the principal quintic via the Brioschi and Rogers-Ramanujan cfrac $R(q)$". Mathematics Stack Exchange.
- ^ Jerrard, George Birch (1859). An essay on the resolution of equations. London, UK: Taylor & Francis.
- ^ Adamchik, Victor (2003). "Polynomial Transformations of Tschirnhaus, Bring, and Jerrard" (PDF). ACM SIGSAM Bulletin. 37 (3): 91. CiteSeerX 10.1.1.10.9463. doi:10.1145/990353.990371. S2CID 53229404. Archived from the original (PDF) on 2009-02-26.