Abstract
In our precedence paper (Diouf, Diakhate, \& Watt, 2013), we show the continuity of the zeros of a univariate polynomial which respect to the coefficients. Here we study the sizes of a polynomial and their bounds. The main originality of this paper is maybe a definition of the measure of a polynomial without any reference to the roots, this leads to a very elementary proof of bounds for the factors of polynomials, a subject which is also revisited here. Most of our proofs are extremely simple and all are quite elementary.
Highlights
In our precedence paper (Diouf, Diakhate, & Watt, 2013), we show the continuity of the zeros of a univariate polynomial which respect to the coefficients
We need a lemma to bound for the roots of polynomials, this result is essentially due to Cauchy (1891)
For the proof of this lemma, see Diouf (2007). This lemma enables us to give many inequalities for the complex roots of a polynomial, most of which are due to Cauchy (1891)
Summary
We consider a field K equipped by a non-trivial absolute value, i.e a map x → |x| from K to R such that. Lemma 1 Let P(X) = Xd + ad−1Xd−1 + · · · + a0 be a polynomial with coefficients in K. For the proof of this lemma, see Diouf (2007) This lemma enables us to give many inequalities for the complex roots of a polynomial, most of which are due to Cauchy (1891). Notice that this is enough to prove this inequality for k = 1, the general case follows by induction. Using the notion of compound matrices Specht (1938) proved the following result, for which we do not reproduce the proof. We just notice that the original proof was given for complex numbers but that it works in the case of a field K with an absolute value. We notice that the proof uses Gerschghorin bounds for the eigenvalues of a matrix and that these bounds still hold for matrices with coefficients in K (the proof being the same)
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