We consider the convergence rate of the alternating projection method for the
nontransversal intersection of a semialgebraic set and a linear subspace. For
such an intersection, the convergence rate is known as sublinear in the worst
case. We study the exact convergence rate for a given semialgebraic set and an
initial point, and investigate when the convergence rate is linear or
sublinear. As a consequence, we show that the exact rates are expressed by
multiplicities of the defining polynomials of the semialgebraic set, or related
power series in the case that the linear subspace is a line, and we also decide
the convergence rate for given data by using elimination theory. Our methods
are also applied to give upper bounds for the case that the linear subspace has
the dimension more than one. The upper bounds are shown to be tight by
obtaining exact convergence rates for a specific semialgebraic set, which
depend on the initial points.Comment: 22 pages, 1 figur