Let \(Y\subset {\mathbb{C}}P^ n\) be an algebraic hypersurface of degree d, having only nondegenerate singular points. - Problem: What is a maximal number \(N_ n(d)\) of nondegenerate singular points which can be on a hypersurface of a degree d? - An answer is known only if \(n=1,2:\) if \(n=1\) then \(N_ 1(d)=[d/2],\) if \(n=2\) then \(N_ 2(d)=d(d-1)/2.\) The first nontrivial case is \(n=3\). In 1906, A. B. Basset ”The maximum number of double points on a surface”, Nature 73, 246 (1906) proved \(N_ 3(d)\leq(d(d-1)^ 2-5-\sqrt{d(d-1)(3d-14)+25})/2.\) In this inequality the estimating number has as asymptotic \(d^ 3/2\), when \(d\to \infty\). Basset’s estimation was improved and generalized in following works, but in all cases the estimating number has as asymptotic \(d^ n/2\), when \(d\to \infty\). In this article it is given an estimation \(N_ n(d)\leq A_ n(d)\) with a new asymptotic of \(A_ n(d)\). Namely, \(A_ n(d)=a_ nd^ n+(lower\quad \deg rees\quad of\quad d)\), and \(a_ 3=23/48, a_ n\sim \sqrt{6/\pi n},\) if \(n\to \infty\). The inequality \(N_ n(d)\leq A_ n(d)\) was conjectured by V. I. Arnol’d. The proofs are based on a theory of mixed Hodge structures in vanishing cohomologies. The estimation is a consequence of a general theorem that a spectrum of a quasihomogeneous critical point of a function is more dense than a sum of spectra of critical points, appearing on the same level in any lower deformation of quasihomogeneous critical point. Further results see in ”Semicontinuity of the singularity spectrum” [Preprint 23, Math. Inst., Univ. Leiden, 1-9 (1983)] by J. Steenbrink.