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In ] and, more specifically, in ], the '''principal form''' of an ] 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 ] of either polynomial is a ] of a root of the other polynomial. In ] and, more specifically, in ], the '''principal form''' of an ] 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 ] of either polynomial is a ] of a root of the other polynomial.


The principal form of a polynomial can be found by applying a suitable ] to the given polynomial. The principal form of a polynomial can be found by applying a suitable ] to the given polynomial.

Revision as of 10:36, 31 December 2024

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 root 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

f ( x ) = x n + a 1 x n − 1 + ⋯ + a n − 1 x + a n {\displaystyle f(x)=x^{n}+a_{1}x^{n-1}+\cdots +a_{n-1}x+a_{n}}

be an irreducible polynomial of degree at least three.

Its principal form is a polynomial

g ( y ) = y n + b 3 y n − 3 + ⋯ + b n − 1 y + b n , {\displaystyle g(y)=y^{n}+b_{3}y^{n-3}+\cdots +b_{n-1}y+b_{n},}

together with a Tschirnhaus transformation of degree two

φ ( x ) = x 2 + α x + β {\displaystyle \varphi (x)=x^{2}+\alpha x+\beta }

such that, if r is a root of f, ϕ ( r ) {\displaystyle \phi (r)} is a root of ⁠ g {\displaystyle g} ⁠.

Expressing that ⁠ g {\displaystyle g} ⁠ does not has terms in ⁠ y n − 1 {\displaystyle y^{n-1}} ⁠ and ⁠ y n − 2 {\displaystyle y^{n-2}} ⁠ leads to a system of two equations in ⁠ α {\displaystyle \alpha } ⁠ and ⁠ β {\displaystyle \beta } ⁠, 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.

Literature

References

  1. Weisstein, Eric W. "Principal Quintic Form". mathworld.wolfram.com.
  2. "The solution to the principal quintic via the Brioschi and Rogers-Ramanujan cfrac $R(q)$". Mathematics Stack Exchange.
  3. Jerrard, George Birch (1859). An essay on the resolution of equations. London, UK: Taylor & Francis.
  4. 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.
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