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

k[x_{1},x_{2},\ldots ,x_{n}]

is a polynomial ring, the ring of radical polynomials is the subring generated by the polynomial^[2]

\sum _{i=1}^{n}x_{i}^{2}.

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). Polynomials. Springer Science & Business Media. ISBN 978-0-387-40627-5.
^ 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.

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[1] Barbeau, E. J. (2003-10-09). Polynomials. Springer Science & Business Media. ISBN 978-0-387-40627-5.

[2] 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.

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