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

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

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

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

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.

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