We compute the persistence for the 2d-diffusion equation with random initial condition, i.e., the probability p0(t) that the diffusion field, at a given point x in the plane, has not changed sign up to time t. For large t, we show that p0(t) ∼ t −θ(2) with θ(2) = 3/16. Using the connection between the 2d-diffusion equation and Kac random polynomials, we show that the probability q0(n) that Kac polynomials, of (even) degree n, have no real root decays, for large n, as q0(n) ∼ n −3/4 . We obtain this result by using yet another connection with the truncated orthogonal ensemble of random matrices. This allows us to compute various properties of the zero-crossings of the diffusing field, equivalently of the real roots of Kac polynomials. Finally, we unveil a precise connection with a fourth model: the semi-infinite Ising spin chain with Glauber dynamics at zero temperature. arXiv:1806.11275v1 [cond-mat.stat-mech]