We are concerned with the Umemura polynomials associated with the third
Painlev\'e equation. We extend Taneda's method, which was developed for the
Yablonskii--Vorob'ev polynomials associated with the second Painlev\'e
equation, to give an algebraic proof that the rational functions generated by
the nonlinear recurrence relation satisfied by Umemura polynomials are indeed
polynomials. Our proof is constructive and gives information about the roots of
the Umemura polynomials.Comment: 20 pages, 3 figure