Radical polynomial
In mathematics, in the realm of abstract algebra, a radical polynomial is a multivariate polynomial[1] over a field that can be expressed as a polynomial in the sum of squares of the variables. That is, if
- [math]\displaystyle{ k[x_1, x_2,\ldots, x_n] }[/math]
is a polynomial ring, the ring of radical polynomials is the subring generated by the polynomial[2]
- [math]\displaystyle{ \sum_{i=1}^n x_i^2. }[/math]
Radical polynomials are characterized as precisely those polynomials that are invariant under the action of the orthogonal group.
The ring of radical polynomials is a graded subalgebra of the ring of all polynomials.
The standard separation of variables theorem asserts that every polynomial can be expressed as a finite sum of terms, each term being a product of a radical polynomial and a harmonic polynomial. This is equivalent to the statement that the ring of all polynomials is a free module over the ring of radical polynomials.
References
- ↑ Barbeau, E. J. (2003-10-09) (in en). Polynomials. Springer Science & Business Media. ISBN 978-0-387-40627-5. https://books.google.com/books?id=CynRMm5qTmQC.
- ↑ Sethuraman, B. A. (1997). Rings, fields, and vector spaces : an introduction to abstract algebra via geometric constructability. Internet Archive. New York : Springer. ISBN 978-0-387-94848-5. http://archive.org/details/ringsfieldsvecto0000seth.
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