Symmetric Polynomial - Relation With The Roots of A Monic Univariate Polynomial

Relation With The Roots of A Monic Univariate Polynomial

Consider a monic polynomial in t of degree n

with coefficients ai in some field k. There exist n roots x1,…,xn of P in some possibly larger field (for instance if k is the field of real numbers, the roots will exist in the field of complex numbers); some of the roots might be equal, but the fact that one has all roots is expressed by the relation

By comparison of the coefficients one finds that

\begin{align} a_{n-1}&=-x_1-x_2-\cdots-x_n\\ a_{n-2}&=x_1x_2+x_1x_3+\cdots+x_2x_3+\cdots+x_{n-1}x_n = \textstyle\sum_{1\leq i<j\leq n}x_ix_j\\ & {}\ \, \vdots\\ a_1&=(-1)^{n-1}(x_2x_3\cdots x_n+x_1x_3x_4\cdots x_n+\cdots+x_1x_2\cdots x_{n-2}x_n+x_1x_2\cdots x_{n-1}) = \textstyle(-1)^{n-1}\sum_{i=1}^n\prod_{j\neq i}x_j\\ a_0&=(-1)^nx_1x_2\cdots x_n.\\ \end{align}

These are in fact just instances of Viète's formulas. They show that all coefficients of the polynomial are given in terms of the roots by a symmetric polynomial expression: although for a given polynomial P there may be qualitative differences between the roots (like lying in the base field k or not, being simple or multiple roots), none of this affects the way the roots occur in these expressions.

Now one may change the point of view, by taking the roots rather than the coefficients as basic parameters for describing P, and considering them as indeterminates rather than as constants in an appropriate field; the coefficients ai then become just the particular symmetric polynomials given by the above equations. Those polynomials, without the sign, are known as the elementary symmetric polynomials in x1,…,xn. A basic fact, known as the fundamental theorem of symmetric polynomials states that any symmetric polynomial in n variables can be given by a polynomial expression in terms of these elementary symmetric polynomials. It follows that any symmetric polynomial expression in the roots of a monic polynomial can be expressed as a polynomial in the coefficients of the polynomial, and in particular that its value lies in the base field k that contains those coefficients. Thus, when working only with such symmetric polynomial expressions in the roots, it is unnecessary to know anything particular about those roots, or to compute in any larger field than k in which those roots may lie. In fact the values of the roots themselves become rather irrelevant, and the necessary relations between coefficients and symmetric polynomial expressions can be found by computations in terms of symmetric polynomials only. An example of such relations are Newton's identities, which express the sum of any fixed power of the roots in terms of the elementary symmetric polynomials.

Read more about this topic:  Symmetric Polynomial

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