Smoothing of real algebraic hypersurfaces by rigid isotopies
Alexander Nabutovsky · Annales de l’institut Fourier · 1991
Define for a smooth compact hypersurface M n of R n + 1 its crumpleness κ ( M n ) as the ratio diam R n + 1 ( M n ) / r ( M n ) , where r ( M n ) is the distance from M n to its central set. (In other words, r ( M n ) is the maximal radius of an open non-selfintersecting tube around M n in R n + 1 . ) We prove that any n -dimensional non-singular compact algebraic hypersurface of degree d is rigidly isotopic to an algebraic hypersurface of degree d and of crumpleness ≤ exp ( c ( n ) d α ( n ) d n + 1 ) . Here c ( n ) , α ( n ) depend only on n , and rigid isotopy means an isotopy passing only through hypersurfaces of degree ≤ d . As an application, we show that for some constants c , β any two isotopic smooth non-singular algebraic compact curves of degree ≤ d in R 2