  In this paper, we propose new linearly convergent second-order methods for
minimizing convex quartic polynomials. This framework is applied for designing
optimization schemes, which can solve general convex problems satisfying a new
condition of quartic regularity. It assumes positive definiteness and
boundedness of the fourth derivative of the objective function. For such
problems, an appropriate quartic regularization of Damped Newton Method has
global linear rate of convergence. We discuss several important consequences of
this result. In particular, it can be used for constructing new second-order
methods in the framework of high-order proximal-point schemes. These methods
have convergence rate $\tilde O(k^{-p})$, where $k$ is the iteration counter,
$p$ is equal to 3, 4, or 5, and tilde indicates the presence of logarithmic
factors in the complexity bounds for the auxiliary problems, which are solved
at each iteration of the schemes.
