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It bounds the entropy of the near critical values of Df0 . Its result can be easily translated into the following form: For any r > 0 there are γ(K, r)-regular values v of Df0 , at a distance at most r from zero. Here γ(K, r) is an explicitely given function, tending to zero as r tends to zero (see [Yom 3] and [Yom 18]). For a given > 0 let us pick a certain γ(K, )-regular value v of Df0 , at a distance at most from zero, and let h be a linear function with Dh = −v. Then all the critical points of f = f0 + h have the Hessian with the minimal eigenvalue bounded from below by γ(K, ).

15. For any R1 < R, y˜ ∈ BR1 ,K , with K = [ Here A is an absolute constant. 161 4A(R + R1 ) 2m+2 ] . 15 provides a structural Bernstein inequality for algebraic functions of one variable. It can be easily extended to algebraic functions of several variables (see [Roy-Yom]). One can restate this theorem (or, rather, its generalization to multivalued functions) in a more geometric way: if an algebraic curve Y of degree d in C 2 is contained, over the disk DR in the x-axis, in a tube of the size K, and it does not blow up to infinity over the disk D3R , then it is contained in a tube of the size C(d)K over the disk D2R .

For any R1 < R, y˜ ∈ BR1 ,K , with K = [ Here A is an absolute constant. 161 4A(R + R1 ) 2m+2 ] . 15 provides a structural Bernstein inequality for algebraic functions of one variable. It can be easily extended to algebraic functions of several variables (see [Roy-Yom]). One can restate this theorem (or, rather, its generalization to multivalued functions) in a more geometric way: if an algebraic curve Y of degree d in C 2 is contained, over the disk DR in the x-axis, in a tube of the size K, and it does not blow up to infinity over the disk D3R , then it is contained in a tube of the size C(d)K over the disk D2R .

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