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In complex analysis, the Gauss-Lucas theorem gives a geometrical relation between the roots of a polynomial P and the roots of its derivative P'. The set of roots of a real or complex polynomial is a set of points in the complex plane. The theorem states that the roots of P' all lie within the convex hull of the roots of P, that is the smallest convex polygon containing the roots of P. When P has a single root then this convex hull is a single point and when the roots lie on a line then the convex hull is a segment of this line. The Gauss-Lucas Theorem, named after Karl Friedrich Gauss and Édouard Lucas is similar in spirit to Rolle's Theorem.

In addition, if a polynomial of degree n of real coefficients has n distinct real zeros $x\_1,\; math>,\; we\; see,\; usingRolle\text{'}s\; theorem,\; that\; the\; zeros\; of\; the\; derivative\; polynomial\; are\; in\; the\; interval$ [x\_1,x\_n],$which\; is\; the\; convex\; hull\; of\; the\; set\; of\; roots.$

- $P(z)=\; alpha\; prod\_\{i=1\}^n\; (z-a\_i)$

where the complex numbers $a\_1,\; a\_2,\; ldots,\; a\_n$ are the – not necessary distinct – zeros of the polynomial $P$, the complex number $alpha$ is the leading coefficient of $P$ and $n$ is the degree of $P$. Let $z$ be any complex number for which $P(z)\; neq\; 0$. Then we have for the Logarithmic derivative

- $frac\{P^prime(z)\}\{P(z)\}=\; sum\_\{i=1\}^n\; frac\{1\}\{z-a\_i\}.$

In particular, if $z$ is a zero of $P\text{'}$ and still $P(z)\; neq\; 0$, then

- $sum\_\{i=1\}^n\; frac\{1\}\{z-a\_i\}=0.$

or

- $sum\_\{i=1\}^n\; frac\{overline\{z\}-overline\{a\_i\}\; \}\; \{vert\; z-a\_ivert^2\}=0.$

This may also be written as

- $left(sum\_\{i=1\}^n\; frac\{1\}\{vert\; z-a\_ivert^2\}right)overline\{z\}=$

Taking their conjugates, we see that z is a weighted sum with positive coefficients that sum to one, or the barycenter, of the complex numbers a_{i} (with different mass assigned on each root).

If P(z) = P'(z) = 0, then z = 1·z + 0·a_{i}, and is still a convex combination of the roots of P.

- Lucas-Gauss Theorem by Bruce Torrence, The Wolfram Demonstrations Project.

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Last updated on Saturday September 20, 2008 at 03:49:10 PDT (GMT -0700)

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Last updated on Saturday September 20, 2008 at 03:49:10 PDT (GMT -0700)

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