Talk:Geometrical properties of polynomial roots

Gauss-Lucas theorem

There is an article on the Gauss-Lucas theorem on the French Wikipedia [1]. Haseldon 19:29, 31 January 2007 (UTC)

Name?

You might consider moving this page to polynomial roots which is currently just a redirect to root (mathematics). -- Fropuff 03:24, 3 February 2007 (UTC)

Positive Hessian

I renoved the following text from the article:

If the polynomial $f$ has real simple roots the Hessian [footnote: In fact, this Hessian is obtained by homogeneizing f, and substituting by 1 the homogenizing variable in the determinant of the Hessian matrix of the resulting polynomial.] $H(f)$ evaluated on the interval [ -1, 1 ] is always ≥ 0.[ref name="Laguerre1880"/] In symbols

$H(f)=(n-1)^{2}f''^{2}-n(n-1)ff'\geq 0$

where f’ is the derivative of f with respect to x, and f’’ is the second derivative.{{verify source}}

This relation applied to polynomials with complex roots is known as Bernstein's inequality.{{cn}}

This was already considered dubious by another user (see history) and when I checked the Laguerre reference, I found that the title is wrong, the article this only in passing ("as we know, H is also positive"), but nothing about the interval [-1,1] and the formula for the Hessian is also wrong. -- Jitse Niesen (talk) 13:19, 3 September 2013 (UTC)

Routh Hurwitz

The Routh Hurwitz criterion seems to give additional information about the real part of roots. Should it be added? — Preceding unsigned comment added by 129.215.90.214 (talk) 11:28, 30 October 2013 (UTC)

Definition of a polynomial

Following discussion between myself and D.Lazard, I am changing the definition of "polynomial" in the lead. The old version says that a polynomial is an expression the form [sic] $p=a_{0}+a_{1}x+\cdots +a_{n}x^{n},$whereas in fact it is an expression of the form $a_{0}+a_{1}x+\cdots +a_{n}x^{n}.$ A mathematician is likely, in the context, to read "an expression p = XYZ " as a short hand for "an expression XYZ, which will be referred to as p for convenience of reference", but most non-mathematicians are likely to read it as meaning exactly what it says, and think that a "polynomial" is a type of equation. The editor who uses the pseudonym "JamesBWatson" (talk) 09:09, 14 May 2014 (UTC)

Cohen, Alan M. Paper

So the bound from the paper Cohen, Alan M. (2009). "Bounds for the roots of polynomial equations". Mathematical Gazette. Cannot possibly be true. Consider x^3-x-1. Here the max is 1 but it clearly has a root larger than one. He probably meant to have a 1+ in front of each term in the max. This is especially clear if you look at the form of the companion matrix and apply Gershgorn. — Preceding unsigned comment added by 98.249.79.241 (talk) 03:04, 12 September 2018 (UTC)

Bounds

The section on bounds of all roots was a mess. I have started to restructuring it. It contained several bounds, some of them being wrongly copied from the source (for example bounds involving quotients of coefficients, that may be zero, of reversion of the indexing the coefficients (typically, $a_{i}^{1/i}$ instead of $a_{i}^{1/(n-i)}$). Also, many bounds are credited to Lagrange, without a clear source for this attribution.

In a first step, I reformulate the description of the bounds for clarifying them. I keep all the bonnds that were given, even if they are always worse than another one, because a global view is needed for being sure that they are never useful. As many of these bounds are not clearly sourced, I have provided a (collapsed) proof when I know of a simple one, and I have tagged the others. In a second step, I'll improve sourcing. If anyone can help me for that, this will be welcome. D.Lazard (talk) 17:03, 6 March 2019 (UTC)

Fujiwara's bound

The source for Fujiwara's bound does not actually seem to give the bound stated on the page:

$2\,\max \left\{\left|{\frac {a_{n-1}}{a_{n}}}\right|,\left|{\frac {a_{n-2}}{a_{n}}}\right|^{\frac {1}{2}},\ldots ,\left|{\frac {a_{1}}{a_{n}}}\right|^{\frac {1}{n-1}},\left|{\frac {a_{0}}{2a_{n}}}\right|^{\frac {1}{n}}\right\}.$

This paper does not mention dividing the last term by 2, instead giving the bound

$2\,\max \left\{\left|{\frac {a_{n-1}}{a_{n}}}\right|,\left|{\frac {a_{n-2}}{a_{n}}}\right|^{\frac {1}{2}},\ldots ,\left|{\frac {a_{1}}{a_{n}}}\right|^{\frac {1}{n-1}},\left|{\frac {a_{0}}{a_{n}}}\right|^{\frac {1}{n}}\right\},$

"originally given by Lagrange, but attributed to Zassenhaus by Donald Knuth" earlier in the page. Lagrange also had an improvement of this bound which replaces "2 max" with the sum of the two largest element (proof given in Maurice Mignotte, Doru Stefanescu. On an estimation of polynomial roots by Lagrange. 2002). Can someone find the right source for the first bound or show me where it explains this in the original source? Also, is it not worth adding this improved Lagrange bound to this page? CellThirtyFour (talk) 13:15, 30 July 2024 (UTC)

References