State and output dead beat controllability tests for a very large class of polynomial systems with rational coefficients may be based on the Quantifier Elimination by Partial Cylindrical Algebraic Decomposition (QEPCAD) symbolic computation program. The method is unified for a very large class of systems and can handle one- or two-sided control constraints, Families of minimum time state/output dead beat controllers are obtained, The computational complexity of the test is prohibitive for general polynomial systems, but by constraining the structure of the system we may beat the curse of complexity. A computationally less expensive algebraic test for output dead beat controllability for a class of odd polynomial systems is presented. Necessary and sufficient conditions are given, They are still very difficult to check, Therefore, a number of easier-to-check sufficient conditions are also provided, The latter are based on the Grobner basis method and QEPCAD, It is shown on a subclass of odd polynomial systems how it is possible to further reduce the computational complexity by exploiting the structure of the system.