  The one-dimensional Schroedinger's equation is analysed with regard to the
existence of exact solutions for decatic polynomial potentials. Under certain
conditions on the potential's parameters, we show that the decatic polynomial
potential $V(x)=ax^{10}+bx^8+cx^6+dx^4+ex^2$, $a>0$ is exactly solvable. By
examining the polynomial solutions of certain linear differential equations
with polynomial coefficients, the necessary and sufficient conditions for
corresponding energy-dependent polynomial solutions are given in detail. It is
also shown that these polynomials satisfy a four-term recurrence relation,
whose real roots are the exact energy eigenvalues. Further, it is shown that
these polynomials generate the eigenfunction solutions of the corresponding
Schroedinger equation. Further analysis for arbitrary values of the potential
parameters using the asymptotic iteration method is also presented.
