Let l linear forms and k quadratic forms (n = l + 2k) be given in the n-dimensional real space R. Absolute values of the forms define a map of the space R into the positive ortant S₊ of the m-dimensional real space S, where m = l + k. Here the integer lattice in R is mapped into a set Z ⊂ S₊. The closure of the convex hull G of the set Z ∖0 is a polyhedral set. Integer points from R, which are mapped in the boundary ∂G of the polyhedron G, give the best Diophantine approximations to root subspaces of all given forms. In the algebraic case, when the given forms are connected with roots of a polynomial of degree n, we prove that the polyhedron G has m− 1 independent periods. It is a generalization of the Lagrange Theorem, that continued fractions of a square irrationality is periodic.

Mathematical problems and theory of numerical methods