Local topological algebraicity of analytic function germs

Transcription

1 Local topological algebraicity of analytic function germs joint work with Marcin Bilski and Guillaume Rond Adam Parusiński Université Nice Sophia Antipolis Singularities in geometry, topology, foliations, and dynamics Mérida, México Celebrating 60th birthday of Pepe Seade Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

2 GDR Singularités et Applications web page for singularity theory Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

3 GDR Singularités et Applications web page for singularity theory Real Analytic Geometry and Trajectories of Vector Fields Luminy, June 8-12, 2015 organizers: K. Kurdyka, A. Parusiński, J.P. Rolin, F. Sanz Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

4 1 Theorem of Mostowski. 2 Statement of results. 3 Artin approximation. 4 Zariski equisingularity. 5 Proof. Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

5 Theorem (T. Mostowski, Bull. Polish Acad. Sci. Math., 1984) Let (V, 0) (R n, 0) be an analytic set germ. Then there is a homeomorphism h : (R m, 0) (R m, 0), m 2n + 1, such that h(v ) is an algebraic set germ. Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

6 Theorem (T. Mostowski, Bull. Polish Acad. Sci. Math., 1984) Let (V, 0) (R n, 0) be an analytic set germ. Then there is a homeomorphism h : (R m, 0) (R m, 0), m 2n + 1, such that h(v ) is an algebraic set germ. Remarks: Mostowski s proof works over C. Increasing the embedding dimension n m is not necessary. Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

7 Theorem (T. Mostowski, Bull. Polish Acad. Sci. Math., 1984) Let (V, 0) (R n, 0) be an analytic set germ. Then there is a homeomorphism h : (R m, 0) (R m, 0), m 2n + 1, such that h(v ) is an algebraic set germ. Remarks: Mostowski s proof works over C. Increasing the embedding dimension n m is not necessary. Indeed, to show the last claim: - Mostowski proved that one can take n = m if one requires h(v ) to be a Nash set germ. (Nash = analytic + satisfying algebraic equations) - Nash set germs are Nash equivalent to algebraic set germs. Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

8 Approximation of Nash germs - Nash set germs are Nash equivalent to algebraic set germs by Theorem (J. Bochnak, W. Kucharz, Crelle, 1984) Let K = R or C. For any ideal I K x 1,..., x n there exists a Nash change of coordinates h : (K n, 0) (K n, 0), such that h (I) is generated by polynomials Here we identify - germs of Nash functions f : (K n, 0) K - algebraic power series K x 1,..., x n, i.e. power series that are algebraic over K[x 1,..., x n ]. Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

9 Approximation of Nash germs - Nash set germs are Nash equivalent to algebraic set germs by Theorem (J. Bochnak, W. Kucharz, Crelle, 1984) Let K = R or C. For any ideal I K x 1,..., x n there exists a Nash change of coordinates h : (K n, 0) (K n, 0), such that h (I) is generated by polynomials Here we identify - germs of Nash functions f : (K n, 0) K - algebraic power series K x 1,..., x n, i.e. power series that are algebraic over K[x 1,..., x n ]. Example: 1 + x = x 1 8 x Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

10 Related results: Any isolated singularity f : (K n, 0) K is analytically equivalent to a polynomial germ [Samuel, 1956]. Let V = V 1... V m (R n, 0) analytic coherent, with each V i is isolated singularity and outside the origin V i are in general position. Then, for each k N, V is C k diffeomorphic to an algebraic set germ [Bochnak-Kucharz 1984]. Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

11 Related results: Any isolated singularity f : (K n, 0) K is analytically equivalent to a polynomial germ [Samuel, 1956]. Let V = V 1... V m (R n, 0) analytic coherent, with each V i is isolated singularity and outside the origin V i are in general position. Then, for each k N, V is C k diffeomorphic to an algebraic set germ [Bochnak-Kucharz 1984]. Analytic hypersurface X (C n, 0) with dim SingX = 1 is homeomorphic to an algebraic hypersurface germ. [J. Fernández de Bobadilla, J. Algebraic Geom., 2012] Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

12 Related results: Any isolated singularity f : (K n, 0) K is analytically equivalent to a polynomial germ [Samuel, 1956]. Let V = V 1... V m (R n, 0) analytic coherent, with each V i is isolated singularity and outside the origin V i are in general position. Then, for each k N, V is C k diffeomorphic to an algebraic set germ [Bochnak-Kucharz 1984]. Analytic hypersurface X (C n, 0) with dim SingX = 1 is homeomorphic to an algebraic hypersurface germ. [J. Fernández de Bobadilla, J. Algebraic Geom., 2012] Examples. The germ (V, 0) (K 3, 0), defined by xy(y x)(y (3 + t)x)(y 4e t x) = 0 is not C 1 -diffeomorphic to an algebraic set germ. [Whitney, 1965] Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

13 Related results: Any isolated singularity f : (K n, 0) K is analytically equivalent to a polynomial germ [Samuel, 1956]. Let V = V 1... V m (R n, 0) analytic coherent, with each V i is isolated singularity and outside the origin V i are in general position. Then, for each k N, V is C k diffeomorphic to an algebraic set germ [Bochnak-Kucharz 1984]. Analytic hypersurface X (C n, 0) with dim SingX = 1 is homeomorphic to an algebraic hypersurface germ. [J. Fernández de Bobadilla, J. Algebraic Geom., 2012] Examples. The germ (V, 0) (K 3, 0), defined by xy(y x)(y (3 + t)x)(y 4e t x) = 0 is not C 1 -diffeomorphic to an algebraic set germ. [Whitney, 1965] The function germ f (x, y, z) = e z xyz(x 4 + y 4 + x 3 yz + x 2 y 3 z) is not analytically equivalent to a polynomial germ. [Pellikaan, 1988] Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

14 Statement of results. Theorem (M. Bilski, G. Rond, A.P., to appear in J. Alg. Geom.) Let g : (K n, 0) (K, 0) be an analytic function germ. Then there is a homeomorphism σ : (K n, 0) (K n, 0) such that g σ is a polynomial germ. Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

15 Statement of results. Theorem (M. Bilski, G. Rond, A.P., to appear in J. Alg. Geom.) Let g : (K n, 0) (K, 0) be an analytic function germ. Then there is a homeomorphism σ : (K n, 0) (K n, 0) such that g σ is a polynomial germ. Theorem ( ) Let, moreover, (V i, 0) (K n, 0) be a finite family of analytic set germs. Then there is σ : (K n, 0) (K n, 0) such that g σ is a polynomial germ, and σ 1 (V i ) are algebraic set germs. Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

16 Structure of proof analytic algebraic (Nash) polynomial Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

17 Structure of proof analytic algebraic (Nash) polynomial 1 Płoski s version of Artin approximation Deformation of an analytic germ to a Nash germ such that this deformation is Zariski equisingular. Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

18 Structure of proof analytic algebraic (Nash) polynomial 1 Płoski s version of Artin approximation Deformation of an analytic germ to a Nash germ such that this deformation is Zariski equisingular. 2 Zariski equisingularity = topological equisingularity (Varchenko, 1972). Conclusion: analytic germs top Nash germs. Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

19 Structure of proof analytic algebraic (Nash) polynomial 1 Płoski s version of Artin approximation Deformation of an analytic germ to a Nash germ such that this deformation is Zariski equisingular. 2 Zariski equisingularity = topological equisingularity (Varchenko, 1972). Conclusion: analytic germs top Nash germs. 3 Approximation of Nash germs by polynomial germs. (Bochnak-Kucharz based on Artin-Mazur) Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

20 Artin Approximation Let x = (x 1,..., x n ), y = (y 1,..., y m ) and suppose there exists a convergent solution y(x) (K{x}) m of f i (x, y(x)) = 0 with f i K x [y], i = 1,..., p. Then there is an algebraic power series solution ŷ(x) (K x ) m. Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

21 Artin Approximation Let x = (x 1,..., x n ), y = (y 1,..., y m ) and suppose there exists a convergent solution y(x) (K{x}) m of f i (x, y(x)) = 0 with f i K x [y], i = 1,..., p. Then there is an algebraic power series solution ŷ(x) (K x ) m. Płoski s conclusion: There exist new variables z = (z 1,..., z s ) and algebraic power series y(x, z) (K x, z ) m, s.t. f i (x, y(x, z)) = 0 i = 1,..., p, Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

22 Artin Approximation Let x = (x 1,..., x n ), y = (y 1,..., y m ) and suppose there exists a convergent solution y(x) (K{x}) m of f i (x, y(x)) = 0 with f i K x [y], i = 1,..., p. Then there is an algebraic power series solution ŷ(x) (K x ) m. Płoski s conclusion: There exist new variables z = (z 1,..., z s ) and algebraic power series y(x, z) (K x, z ) m, s.t. and z i (x) K{x}, z i (0) = 0, s.t. f i (x, y(x, z)) = 0 i = 1,..., p, y(x) = y(x, z(x)). Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

23 Nested Artin-Płoski-Popescu Approximation Corollary of Spivakovsky s proof of the Néron-Popescu Desingularization. Theorem Suppose there exists a solution y(x) (K{x}) m of f i (x, y(x)) = 0 with f i K x [y], i = 1,..., p. s.t. y i (x) depends only on (x 1,..., x σ(i) ), i σ(i) increasing. Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

24 Nested Artin-Płoski-Popescu Approximation Corollary of Spivakovsky s proof of the Néron-Popescu Desingularization. Theorem Suppose there exists a solution y(x) (K{x}) m of f i (x, y(x)) = 0 with f i K x [y], i = 1,..., p. s.t. y i (x) depends only on (x 1,..., x σ(i) ), i σ(i) increasing. Conclusion: There exist z = (z 1,..., z s ) and y(x, z) (K x, z ) m, s.t. f i (x, y(x, z)) = 0 i = 1,..., p, with y i (x, z) K x 1,..., x σ(i), z 1,..., z τ(i). Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

25 Nested Artin-Płoski-Popescu Approximation Corollary of Spivakovsky s proof of the Néron-Popescu Desingularization. Theorem Suppose there exists a solution y(x) (K{x}) m of f i (x, y(x)) = 0 with f i K x [y], i = 1,..., p. s.t. y i (x) depends only on (x 1,..., x σ(i) ), i σ(i) increasing. Conclusion: There exist z = (z 1,..., z s ) and y(x, z) (K x, z ) m, s.t. f i (x, y(x, z)) = 0 i = 1,..., p, with y i (x, z) K x 1,..., x σ(i), z 1,..., z τ(i). There are z i (x) K{x}, z i (0) = 0, with z 1 (x),..., z τ(i) (x) depending only on (x 1,..., x σ(i) ), i τ(i) increasing, s.t. y(x) = y(x, z(x)). Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

26 Puiseux with parameter Suppose given complex analytic germ F (t, x, z) = z p + p a j (t, x)z p j x C, z C, a j C{t, x}, a j (t, 0) 0, where t C l parameter. Assumption: The discriminant of F is of the form Theorem j=1 F (t, x) = x M unit(t, x) Then there is d N and ã j (t, y) C{t, y} s.t. F (t, y d, z) = p (z ã j (t, y)). j=1 Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

27 Puiseux with parameter Suppose given complex analytic germ F (t, x, z) = z p + p a j (t, x)z p j x C, z C, a j C{t, x}, a j (t, 0) 0, where t C l parameter. Assumption: The discriminant of F is of the form Theorem j=1 F (t, x) = x M unit(t, x) Then there is d N and ã j (t, y) C{t, y} s.t. Corollary F (t, y d, z) = p (z ã j (t, y)). j=1 Such a family of plane curve germs: t {F (t, x, z) = 0} is topologically trivial. Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

28 Puiseux with parameter F (t, x, z) = z p + p a j (t, x)z p j, F (t, x) = x M unit(t, x) j=1 The family of plane curve germs: t V t (F ) = {F (t, x, y) = 0} is ambient topologically trivial, that is: There exist neighborhoods 0 B C l, 0 Ω 0 C 2 and 0 Ω C l C 2, and a homeomorphism h(t, x, z) : B Ω 0 Ω, h(t, x, z) = (t, x, ψ(t, x, z)) h(t, 0) = (t, 0) and h(0, x) = (0, x) preserving the zero set of F : h(b V 0 (F )) = V (F ). Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

29 Zariski equisingularity Suppose given a system of complex analytic germs F i (t, x i ) = x p i p i i + j=1 a i 1,j (t, x i 1 )x p i j i, i = 0,..., n, t C l, x i = (x 1,..., x i ) C i, a i 1,j C{t, x i 1 }, a i 1,j (t, 0) 0, s.t. Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

30 Zariski equisingularity Suppose given a system of complex analytic germs F i (t, x i ) = x p i p i i + j=1 a i 1,j (t, x i 1 )x p i j i, i = 0,..., n, t C l, x i = (x 1,..., x i ) C i, a i 1,j C{t, x i 1 }, a i 1,j (t, 0) 0, s.t. This family is Zariski equisingular along T = C l {0} at the origin if 1 F i 1 (t, x i 1 ) = 0 is the Weierstrass polynomial associated to the discriminant (F i,red ) of F i reduced. 2 There is i, s.t. F i (0) 1 (then we set F k 1 for all k i). Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

31 Zariski equisingularity Suppose given a system of complex analytic germs F i (t, x i ) = x p i p i i + j=1 a i 1,j (t, x i 1 )x p i j i, i = 0,..., n, t C l, x i = (x 1,..., x i ) C i, a i 1,j C{t, x i 1 }, a i 1,j (t, 0) 0, s.t. This family is Zariski equisingular along T = C l {0} at the origin if 1 F i 1 (t, x i 1 ) = 0 is the Weierstrass polynomial associated to the discriminant (F i,red ) of F i reduced. 2 There is i, s.t. F i (0) 1 (then we set F k 1 for all k i). Theorem (Varchenko, 1972) If F (t, x) can be completed (F n = F ) to a Zariski equisingular family (maybe after a local change of coordinates) then the family of set germs t {F (t, x) = 0} is ambient topologically trivial. Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

32 Zariski equisingularity implies topological triviality Theorem (Varchenko, 1972) If F (t, x) can be completed (F n = F ) to a Zariski equisingular family (maybe after a local change of coordinates) then the family of set germs t V t (F ) = {F (t, x, y) = 0} is ambient topologically trivial. There exist neighborhoods 0 B C l, 0 Ω 0 C n and 0 Ω C l C n, and a homeomorphism h(t, x) : B Ω 0 Ω, h(t, x) = (t, h 1 (t, x 1 ), h 2 (t, x 2 ),..., h n (t, x n )) h(t, 0) = (t, 0) and h(0, x) = (0, x) preserving the zero set of F : h(t V 0 (F )) = V (F ). Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

33 Varchenko s construction. Remarks. K = R: If the coefficients of F i s are real the whole construction can be chosen conjugation invariant. one can always choose h 1 (t, x 1 ) = x 1. Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

34 Varchenko s construction. Remarks. K = R: If the coefficients of F i s are real the whole construction can be chosen conjugation invariant. one can always choose h 1 (t, x 1 ) = x 1. can be adapted to the families of function germs g t (x 2,..., x n ) as follows. Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

35 Varchenko s construction. Remarks. K = R: If the coefficients of F i s are real the whole construction can be chosen conjugation invariant. one can always choose h 1 (t, x 1 ) = x 1. can be adapted to the families of function germs g t (x 2,..., x n ) as follows. Define F (t, x) = x1 g t(x 2,..., x n). Consider systems of function Fi (t, x i ) with relaxed condition: There are q i N s.t. the discriminants of F i,red (t, x i ) equals x q i 1 i 1 (t, x i 1 ) and F i 1 is the Weierstrass polynomial associated to i 1. Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

36 Varchenko s construction. Remarks. K = R: If the coefficients of F i s are real the whole construction can be chosen conjugation invariant. one can always choose h 1 (t, x 1 ) = x 1. can be adapted to the families of function germs g t (x 2,..., x n ) as follows. Define F (t, x) = x1 g t(x 2,..., x n). Consider systems of function Fi (t, x i ) with relaxed condition: There are q i N s.t. the discriminants of F i,red (t, x i ) equals x q i 1 i 1 (t, x i 1 ) and F i 1 is the Weierstrass polynomial associated to i 1. If this system is Zariski equisingular then gt can by trivialized :: g 0 = g t σ t, where if we write h(t, x) = (t, x 1, ψ(t, x)) then σ t(y) = ψ(t, g 0(y), y). Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

37 Given f K{x}. We construct f n (x), f n 1 (x n 1 ), f n 2 (x n 2 ),..., allowing changes of local coordinates if necessary. Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

38 Given f K{x}. We construct f n (x), f n 1 (x n 1 ), f n 2 (x n 2 ),..., allowing changes of local coordinates if necessary. f n is the Weierstrass polynomial associated to f : f n (x) = x pn p n n + j=1 a n 1,j (x n 1 )x pn j n. Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

39 Given f K{x}. We construct f n (x), f n 1 (x n 1 ), f n 2 (x n 2 ),..., allowing changes of local coordinates if necessary. f n is the Weierstrass polynomial associated to f : f n (x) = x pn p n n + j=1 a n 1,j (x n 1 )x pn j n. Denote a n 1 = (a n 1,1,..., a n 1,pn ). If f n is reduced then, after a linear change of coordinates x n 1, we write the discriminant of f n n (a n 1 ) = u n 1 (x n 1 )(x pn 1 n 1 pn 1 + j=1 a n 2,j (x n 2 )x pn 1 j n 1 ). (If f n is not reduced we use the generalized discriminants). f n 1 (x n 1 ) := x pn 1 n 1 pn 1 + j=1 a n 2,j (x n 2 )x pn 1 j n 1 Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

40 Given f K{x}. We construct f n (x), f n 1 (x n 1 ), f n 2 (x n 2 ),..., allowing changes of local coordinates if necessary. f n is the Weierstrass polynomial associated to f : f n (x) = x pn p n n + j=1 a n 1,j (x n 1 )x pn j n. Denote a n 1 = (a n 1,1,..., a n 1,pn ). If f n is reduced then, after a linear change of coordinates x n 1, we write the discriminant of f n n (a n 1 ) = u n 1 (x n 1 )(x pn 1 n 1 pn 1 + j=1 a n 2,j (x n 2 )x pn 1 j n 1 ). (If f n is not reduced we use the generalized discriminants). f n 1 (x n 1 ) := x pn 1 n 1 pn 1 + j=1 a n 2,j (x n 2 )x pn 1 j n 1 Then consider the discriminant n 1 (a n 2 ) of f n 1 and continue... Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

41 Consider a system of equations on a i (x i ), u i (x i ): n (a n 1 ) = u n 1 (x n 1 )(x pn 1 n 1 n 1 (a n 2 ) = u n 2 (x n 2 )(x pn 1 n 1 i0 (a i0 1) = u i0 (x i0 ) pn 1 + j=1 pn 1 + a n 2,j (x n 2 )x pn 1 j n 1 ) j=1 a n 2,j (x n 2 )x pn 1 j n 1 ) Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

42 Consider a system of equations on a i (x i ), u i (x i ): n (a n 1 ) = u n 1 (x n 1 )(x pn 1 n 1 n 1 (a n 2 ) = u n 2 (x n 2 )(x pn 1 n 1 i0 (a i0 1) = u i0 (x i0 ) pn 1 + j=1 pn 1 + a n 2,j (x n 2 )x pn 1 j n 1 ) j=1 a n 2,j (x n 2 )x pn 1 j n 1 ) By nested Artin-Płoski-Popescu Approximation there are solutions a i (x i, z) K x i, z, u i (x i, z) K x i, z. and convergent power series z i (x) so that the old solutions can be recover by a i (x i ) = a i (x i, z τ(i) (x i )), u i (x i ) = u i (x i,, z τ(i) (x i )) Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

43 Consider a system of equations on a i (x i ), u i (x i ): n (a n 1 ) = u n 1 (x n 1 )(x pn 1 n 1 n 1 (a n 2 ) = u n 2 (x n 2 )(x pn 1 n 1 i0 (a i0 1) = u i0 (x i0 ) pn 1 + j=1 pn 1 + a n 2,j (x n 2 )x pn 1 j n 1 ) j=1 a n 2,j (x n 2 )x pn 1 j n 1 ) By nested Artin-Płoski-Popescu Approximation there are solutions a i (x i, z) K x i, z, u i (x i, z) K x i, z. and convergent power series z i (x) so that the old solutions can be recover by Define F i (t, x) = x p i i a i (x i ) = a i (x i, z τ(i) (x i )), u i (x i ) = u i (x i,, z τ(i) (x i )) + p i j=1 a i 1,j(x i 1, tz(x i 1 ))x p i j i Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

44 The family F i (t, x) = x p i p i i + j=1 a i 1,j (x i 1, tz(x i 1 )x p i j i, i = 0,..., n 1. is Zariski equisingular (for all t C) and gives a topological equivalence between: V (f ) = V (F n (1, x)) top V (F n (0, x)) the latter being the zero set of a Nash germ F n (0, x) C x. Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19

45 Sto lat Pepe and many happy returns! Adam Parusiński (Nice) Analytic top Algebraic Merida, 8-20 XII, / 19