Estimating the root count,which is the total number of geometrically isolated solutions,of a polynomial system is not only a fundamental study theme in algebraic geometry but also an important subproblem of homotopy continuation methods for solving polynomial systems.For the mixed trigonometric polynomial systems,which are more general than polynomial systems and rather frequently occur in many applications,the classical B′ezout number and the multihomogeneous B′ezout number are the best known upper bounds on the root count.However,for the deficient mixed trigonometric polynomial systems arising in practice,all these upper bounds are far greater than the actual root count.The BKK bound is known as the most accurate upper bound on the root count of polynomial systems.However,the extension of the definition of the BKK bound allowing it to treat mixed trigonometric polynomial systems is very difficult due to the existence of sine and cosine functions.In this paper,two new upper bounds on the root count of a mixed trigonometric polynomial system are defined and the corresponding efficient algorithms for calculating them are also presented.Numerical tests are also given to show the accuracy of these two definitions,and numerically prove they can provide tighter upper bounds on the root count of a mixed trigonometric polynomial system than the existent upper bounds,and also we compare the computational time for calculating these two upper bounds.