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This procedure is implemented as:
int Cauchy_First_Bound_Interval(int Degree,VECTOR &Coeff,INTERVAL &Bound);
with: This procedure returns 0 if Degree=0, Coeff(1)=0 or Coeff(Degree+1)=0 and Degree=1. On success the return code is 1. There is also an implementation for interval polynomial:
int Cauchy_First_Bound_Interval(int Degree,INTERVAL_VECTOR &Coeff,INTERVAL &Bound);
This procedure returns an interval $(\underline{Bound},\overline{Bound})$ such that for any polynomial in the set and for any root $x$ of this polynomial $\underline{Bound} \le \vert x\vert \le \overline{Bound}$. A failure code of 0 is returned if Coeff(1) or Coeff(Degree+1) contains 0 or if Degree=0. On success the returned code is 1.

Jean-Pierre Merlet 2012-12-20