This paper is devoted to the explicit construction of optimal designs for discrimination between two polynomial regression models of degree $n-2$ and $n$. In a fundamental paper, Atkinson and Fedorov [Biometrika 62 (1975a) 57--70] proposed the $T$-optimality criterion for this purpose. Recently, Atkinson [MODA 9, Advances in Model-Oriented Design and Analysis (2010) 9--16] determined $T$-optimal designs for polynomials up to degree 6 numerically and based on these results he conjectured that the support points of the optimal design are cosines of the angles that divide half of the circle into equal parts if the coefficient of $x^{n-1}$ in the polynomial of larger degree vanishes. In the present paper we give a strong justification of the conjecture and determine all $T$-optimal designs explicitly for any degree $n\in\mathbb{N}$. In particular, we show that there exists a one-dimensional class of $T$-optimal designs. Moreover, we also present a generalization to the case when the ratio between the coefficients of $x^{n-1}$ and $x^n$ is smaller than a certain critical value. Because of the complexity of the optimization problem, $T$-optimal designs have only been determined numerically so far, and this paper provides the first explicit solution of the $T$-optimal design problem since its introduction by Atkinson and Fedorov [Biometrika 62 (1975a) 57--70]. Finally, for the remaining cases (where the ratio of coefficients is larger than the critical value), we propose a numerical procedure to calculate the $T$-optimal designs. The results are also illustrated in an example.