Upper Bound Estimation Of Isolated Solution Number Of Mixed Trigonometric Polynomial Systems

The estimation of the number of isolated solutions plays an important role in finding all isolated solutions of polynomial systems by homotopy continuation method. For the mixed trigonometric polynomial systems, the total degree, the multi-homogeneous Bezout number and the generalized Bezout number are known as three upper bounds of the number of isolated solutions. However, for the highly deficient mixed trigonometric polynomial systems arising in the practice, the above bounds are far bigger than the actual number of the isolated solutions.Corresponding to the well known BKK bound, which is the most accurate bound of the iso-lated solution number of polynomial systems, in this paper, we give the BKK bound and quasi BKK bound of mixed trigonometric polynomial systems. At first, we give the BKK bound of mixed trigonometric polynomial systems. We decompose the polynomial system transformed from mixed trigonometric polynomial system, and then calculate the BKK bound of mixed trigonometric polynomial system efficiently by calculating the BKK bound of the tranformed polynomial system. After that, we give the quasi BKK bound of mixed trigonometric polyno-mial systems and its calculation algorithm. Comparing to the calculation algorithm of BKK bound, the calculation of quasi BKK bound only needs to calculate the BKK bound of a poly-nomial system in lower dimensional space. In the end, by numerical tests, we prove that BKK bound and quasi BKK bound are both much tighter upper bound of the solution number of mixed trigonometric polynomial systems. Numerical tests also prove that the calculation algorithm of quasi BKK bound is more efficient than the calculation algorithm of BKK bound, but the quasi BKK bound is maybe less tighter than BKK bound in some cases.