Piers Bohl

In 1908, Bohl established a general theorem for locating the roots of complex trinomials of the form ${\displaystyle P(z)=z^{k}+a\,z^{\ell }+b}$, where ${\displaystyle k}$ and ${\displaystyle \ell }$ are positive integers with ${\displaystyle k>\ell }$, and ${\displaystyle a}$ and ${\displaystyle b}$ are nonzero complex numbers. Rather than relying on heavy algebraic manipulations, he employed an elementary geometric construction: by interpreting the magnitudes of the coefficients ${\displaystyle |a|}$, ${\displaystyle |b|}$ and the chosen radius (for instance, the unit circle) as the sides of a triangle, one can associate two angles that, together with the arguments of ${\displaystyle a}$ and ${\displaystyle b}$, yield explicit bounds. These bounds determine exactly how many roots lie inside the circle, either by simple inequalities when one coefficient dominates, or by counting the integers in a specific interval when all three lengths can form a triangle.^[2]

Bohl's result not only unifies numerous special‐case criteria (such as those later attributed to Schur, Cohn or Jury) but also provides direct formulas that apply regardless of the relative sizes or orientations of the coefficients. Although his work went largely unnoticed for many decades, it anticipates modern applications in the stability analysis of differential and difference equations, where knowing whether all characteristic roots lie within the unit circle is essential for determining asymptotic behaviour.^[2]

• Piers Bohl at the Mathematics Genealogy Project
• Bohl biography at www-history.mcs.st-and.ac.uk
• http://www.mathematics.lv/lms_10_years_after.pdf