# Sketch of Second Proof (after Cauchy)

Consider the quantity *s(z)* = |*P(z)*|, where

P(z) = a_{n}z^{ n} + a_{n -1}z^{ n -1} + ... + a_{1}z^{1} + a_{0}

is a polynomial of degree *n*. Clearly, *s(z)*≠0. If *s(z)* assumes a global minimum value, say, σ, on **C**, then it suffices to prove that σ = 0, or equivalently, it suffices to derive a contradiction from the alternative assumption that σ > 0.

Suppose, then, that *s(z _{0})* = σ > 0. It is convenient to "re-center" the arithmetic by setting

*Q(z)*=

*P(z + z*, so |

_{0})*Q*(0)| = |

*P(z*| = σ, where σ is likewise a global minimum for |

_{0})*Q(z)*|.

Expanding out the definition of |*Q(z)*|, we obtain a set of new coefficients:
*Q(z)* = *b _{n}z^{ n} + b_{n -1}z^{ n -1} + ... + b_{1}z^{1} + b_{0}*, where

*b*=

_{n}*a*≠0, and

_{n}*b*=

_{0}*P(z*≠0. Let

_{0})*m*be the exponent of the lowest power of

*z*in

*Q(z)*whose coefficient

*b*is not zero. Now consider the behavior of

_{m}*Q(z)*for points

*z*whose absolute value is very small, say the points

*z*= lying on a small circle centered at the origin of radius ρ. As observed above, as

*z*sweeps once around this small circle,

*Q(z)*closely approximates the behavior of

*b*, which sweeps out a small circle (of radius |

_{m}z^{m}+ b_{0}*b*|ρ

_{m}^{m}) around the point

*b*=

_{0}*P(z*. (In fact,

_{0})*Q(z)*sweeps around this circle

*m*times.) By choosing ρ sufficiently small, we may ensure that the radius of this circle (that is, the magnitude of

*Q(z) - b*) is smaller than σ. Such a circle will necessarily intersect the line segment connecting the origin to the point

_{0}*b*=

_{0}*P(z*, at a point, say

_{0})*Q(z*,

_{1})*nearer the origin*than

*b*=

_{0}*P(z*. But then |

_{0})*Q(z*| = |

_{1})*P(z*| <σ, contradicting our choice of σ as a global minimum of

_{1}+ z_{0})*s(z)*= |

*P(z)*|. (Actually,

*Q(z)*may only lie near this circle, not on it. Fortunately, the discrepancy consists of the remaining terms of

*Q(z)*(if any), all of which contain powers of ρ higher than

*m*. By taking ρ even smaller, if necessary, we can guarantee that this discrepancy does not wreck the geometry - see Figure 5.)

**Figure 5.**Behavior of

*Q(z)*for "small" values of

*z*. If

*z*lies on the small circle of radius ρ centered at the origin of the

*z*-plane, then

*Q(z)*≈

*b*, which is the larger circle in the figure, centered at

_{m}z^{m}+ b_{0}*Q*(0) =

*b*

_{0}, with radius |

*b*|ρ

^{m}^{m}. For ρ sufficiently small, the remaining terms of (if any) of

*Q(z)*sum to a vector whose magnitude (proportional to ρ

^{n},

*n*>

*m*, is less than the radius of the smaller circle in the figure.) As

*z*sweeps out the small circle of radius ρ on the

*z*-plane, the locus of

*Q(z)*loops around

*Q*(0) =

*b*

_{0}(in fact,

*m*times), and necessarily intersects the vector drawn from the origin of the

*w*-plane to

*Q*(0).

Thus the assumption that σ > 0 is untenable, and we conclude that σ = 0,
that is, that *P(z)* has a complex root.

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Copyright © 1996-2007 Alexander Bogomolny