Algebraic approximation of semianalytic sets

Transcription

1 Algebraic approximation of semianalytic sets M. Ferrarotti, E. Fortuna and L. Wilson Politecnico di Torino Università di Pisa University of Hawai i at Mānoa Liverpool, June 2012

2 An algebraic set in R n is defined by finitely many polynomial equalities.

3 An algebraic set in R n is defined by finitely many polynomial equalities. A semialgebraic set is defined by finite unions of sets defined by finitely many polynomial equalities and inequalities.

4 An algebraic set in R n is defined by finitely many polynomial equalities. A semialgebraic set is defined by finite unions of sets defined by finitely many polynomial equalities and inequalities. A subset of R n is a cone (with respect to 0) if it is closed under multiplication by positive reals.

5 An algebraic set in R n is defined by finitely many polynomial equalities. A semialgebraic set is defined by finite unions of sets defined by finitely many polynomial equalities and inequalities. A subset of R n is a cone (with respect to 0) if it is closed under multiplication by positive reals. The tangent cone to set A at 0 is C(A) = {t p : t 0, p n / p n p, p n A, p n 0 as n }

6 An algebraic set in R n is defined by finitely many polynomial equalities. A semialgebraic set is defined by finite unions of sets defined by finitely many polynomial equalities and inequalities. A subset of R n is a cone (with respect to 0) if it is closed under multiplication by positive reals. The tangent cone to set A at 0 is C(A) = {t p : t 0, p n / p n p, p n A, p n 0 as n } The point 0 can be replaced by any other point in the obvious way.

7 An algebraic set in R n is defined by finitely many polynomial equalities. A semialgebraic set is defined by finite unions of sets defined by finitely many polynomial equalities and inequalities. A subset of R n is a cone (with respect to 0) if it is closed under multiplication by positive reals. The tangent cone to set A at 0 is C(A) = {t p : t 0, p n / p n p, p n A, p n 0 as n } The point 0 can be replaced by any other point in the obvious way. We will henceforth drop the word tangent.

8 Example C(x 2 y 3 = 0)

9 Recall that the cone of an algebraic set is semialgebraic. Example C(x 2 y 3 = 0) = {x = 0, y 0}

10 Recall that the cone of an algebraic set is semialgebraic. Example C(x 2 y 3 = 0) = {x = 0, y 0} FFW, 2000, Pacific J. Math. Every semialgebraic cone of codimension at least 1 is the tangent cone of an algebraic set

11 Example: fit cusp surface to half-plane

12 Example continued: use the cusp trick again C((x 2 y 3 ) 2 z 7 = 0) = {x = 0, y 0, z 0}

13 Generalization of first result Our equivalence relation Two sets A and B are Hausdorff s-equivalent at x (written A s B) if the Pompeiu-Hausdorff distance H(A S r, B S r ) = o(r s ), where S r is the sphere of radius r centered at x.

14 Generalization of first result Our equivalence relation Two sets A and B are Hausdorff s-equivalent at x (written A s B) if the Pompeiu-Hausdorff distance H(A S r, B S r ) = o(r s ), where S r is the sphere of radius r centered at x. Relation to cones For any subanalytic set A and its tangent cone C(A) at x, A 1 C(A). So s gives higher order tangency of sets.

15 Generalization of first result Our equivalence relation Two sets A and B are Hausdorff s-equivalent at x (written A s B) if the Pompeiu-Hausdorff distance H(A S r, B S r ) = o(r s ), where S r is the sphere of radius r centered at x. Relation to cones For any subanalytic set A and its tangent cone C(A) at x, A 1 C(A). So s gives higher order tangency of sets. FFW, 2002, Ann. Scuola Norm. Sup. Pisa Every semialgebraic set of codimension at least 1 is, for every s 1, s-equivalent to an algebraic set

16 What if A is semianalytic or subanalytic? A closed, semianalytic

17 What if A is semianalytic or subanalytic? A closed, semianalytic C(A) is semianalytic

18 What if A is semianalytic or subanalytic? A closed, semianalytic C(A) is semianalytic C(A) is semialgebraic (Hironaka Oslo notes)

19 What if A is semianalytic or subanalytic? A closed, semianalytic C(A) is semianalytic C(A) is semialgebraic (Hironaka Oslo notes) So A 1 C(A) 1 an algebraic set.

20 What if A is semianalytic or subanalytic? A closed, semianalytic C(A) is semianalytic C(A) is semialgebraic (Hironaka Oslo notes) So A 1 C(A) 1 an algebraic set. What if A is subanalytic? Consider A defined by y = sin x, 0 x π, z = 1

21 What if A is semianalytic or subanalytic? A closed, semianalytic C(A) is semianalytic C(A) is semialgebraic (Hironaka Oslo notes) So A 1 C(A) 1 an algebraic set. What if A is subanalytic? Consider A defined by y = sin x, 0 x π, z = 1 A is semianalytic but not semialgebraic

22 What if A is semianalytic or subanalytic? A closed, semianalytic C(A) is semianalytic C(A) is semialgebraic (Hironaka Oslo notes) So A 1 C(A) 1 an algebraic set. What if A is subanalytic? Consider A defined by y = sin x, 0 x π, z = 1 A is semianalytic but not semialgebraic A = C(A, 0) is subanalytic but not semialgebraic since A {z = 1} isn t

23 What if A is semianalytic or subanalytic? A closed, semianalytic C(A) is semianalytic C(A) is semialgebraic (Hironaka Oslo notes) So A 1 C(A) 1 an algebraic set. What if A is subanalytic? Consider A defined by y = sin x, 0 x π, z = 1 A is semianalytic but not semialgebraic A = C(A, 0) is subanalytic but not semialgebraic since A {z = 1} isn t Suppose a semialgebraic B 1 A; then C(B) = A

24 What if A is semianalytic or subanalytic? A closed, semianalytic C(A) is semianalytic C(A) is semialgebraic (Hironaka Oslo notes) So A 1 C(A) 1 an algebraic set. What if A is subanalytic? Consider A defined by y = sin x, 0 x π, z = 1 A is semianalytic but not semialgebraic A = C(A, 0) is subanalytic but not semialgebraic since A {z = 1} isn t Suppose a semialgebraic B 1 A; then C(B) = A B semialgebraic implies so is A; contradiction!

25 Algebraic s > 1 approximation for analytic sets FFW, 2010, Proceedings AMS Assume V = F 1 (0), df surjective at a dense subset of V. Then for every s 1 V is s-equivalent to an algebraic set.

26 Algebraic s > 1 approximation for analytic sets FFW, 2010, Proceedings AMS Assume V = F 1 (0), df surjective at a dense subset of V. Then for every s 1 V is s-equivalent to an algebraic set. Principal methods of proof; inspired by jet sufficiency arguments

27 Algebraic s > 1 approximation for analytic sets FFW, 2010, Proceedings AMS Assume V = F 1 (0), df surjective at a dense subset of V. Then for every s 1 V is s-equivalent to an algebraic set. Principal methods of proof; inspired by jet sufficiency arguments Let G be a sufficiently high degree Taylor polynomial of F

28 Algebraic s > 1 approximation for analytic sets FFW, 2010, Proceedings AMS Assume V = F 1 (0), df surjective at a dense subset of V. Then for every s 1 V is s-equivalent to an algebraic set. Principal methods of proof; inspired by jet sufficiency arguments Let G be a sufficiently high degree Taylor polynomial of F Łojasiewicz inequality F x l off H t (V ) = horn neighborhood t > s implies:

29 Algebraic s > 1 approximation for analytic sets FFW, 2010, Proceedings AMS Assume V = F 1 (0), df surjective at a dense subset of V. Then for every s 1 V is s-equivalent to an algebraic set. Principal methods of proof; inspired by jet sufficiency arguments Let G be a sufficiently high degree Taylor polynomial of F Łojasiewicz inequality F x l off H t (V ) = horn neighborhood t > s implies: V (G) H t (V ), which implies V (G) s V

30 methods continued If df is surjective at x V, G maps some y near x to 0; so V (G) s V

31 Work of M. Bilski We learned in 2009 that Bilski had proved similar results for complex analytic sets V : For any such V and for every s 1, V is s-equivalent to an algebraic set.

32 Work of M. Bilski We learned in 2009 that Bilski had proved similar results for complex analytic sets V : For any such V and for every s 1, V is s-equivalent to an algebraic set. The local multiplicities agree for the analytic and algebraic sets

33 Work of M. Bilski We learned in 2009 that Bilski had proved similar results for complex analytic sets V : For any such V and for every s 1, V is s-equivalent to an algebraic set. The local multiplicities agree for the analytic and algebraic sets The method of proof is very different-does the real case follow?

34 Work of M. Bilski We learned in 2009 that Bilski had proved similar results for complex analytic sets V : For any such V and for every s 1, V is s-equivalent to an algebraic set. The local multiplicities agree for the analytic and algebraic sets The method of proof is very different-does the real case follow? Bilski and Rusek (2010): Nash approximation of analytic sets definable in an o-minimal structure

35 Semianalytic sets FFW, 2012, preliminary Assume A is a closed semi analytic set of codimension at least 1. Then, for every s 1, A is s-equivalent to an algebraic set.

36 Semianalytic sets FFW, 2012, preliminary Assume A is a closed semi analytic set of codimension at least 1. Then, for every s 1, A is s-equivalent to an algebraic set. Principal additional methods of proof

37 Semianalytic sets FFW, 2012, preliminary Assume A is a closed semi analytic set of codimension at least 1. Then, for every s 1, A is s-equivalent to an algebraic set. Principal additional methods of proof We can assume A = {F = 0} {g 1,..., g l 0} = closure of {F = 0} {g 1,..., g l > 0}, and the components of F generate the ideal of V (F)

38 Semianalytic sets FFW, 2012, preliminary Assume A is a closed semi analytic set of codimension at least 1. Then, for every s 1, A is s-equivalent to an algebraic set. Principal additional methods of proof We can assume A = {F = 0} {g 1,..., g l 0} = closure of {F = 0} {g 1,..., g l > 0}, and the components of F generate the ideal of V (F) If V (F) is not a complete intersection, we can find a projection π such that F = π F satisfies

39 Semianalytic sets FFW, 2012, preliminary Assume A is a closed semi analytic set of codimension at least 1. Then, for every s 1, A is s-equivalent to an algebraic set. Principal additional methods of proof We can assume A = {F = 0} {g 1,..., g l 0} = closure of {F = 0} {g 1,..., g l > 0}, and the components of F generate the ideal of V (F) If V (F) is not a complete intersection, we can find a projection π such that F = π F satisfies df is generically surjective on V (F ) and dim V (F ) = dim V (F )

40 Semianalytic sets FFW, 2012, preliminary Assume A is a closed semi analytic set of codimension at least 1. Then, for every s 1, A is s-equivalent to an algebraic set. Principal additional methods of proof We can assume A = {F = 0} {g 1,..., g l 0} = closure of {F = 0} {g 1,..., g l > 0}, and the components of F generate the ideal of V (F) If V (F) is not a complete intersection, we can find a projection π such that F = π F satisfies df is generically surjective on V (F ) and dim V (F ) = dim V (F ) V (F ) = V (F) W, which are unions of irreducible components of V (F )

41 methods continued A is s-equivalent to V (F ) {g 1,..., g l 0} U, where U = { x 2m F 2 0}

42 methods continued A is s-equivalent to V (F ) {g 1,..., g l 0} U, where U = { x 2m F 2 0}

43 methods continued A is s-equivalent to V (F ) {g 1,..., g l 0} U, where U = { x 2m F 2 0} Induction on dimension of A