New PDF release: A conjecture in arithmetic theory of differential equations

By Katz N.M.

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The intersection number (A 0 b) of the chains A and b is called the linking number of the cycles a and b and denoted l(a, b). Another method of calculating the linking number goes as follows: Let D 2n be the ball, the boundary of which is the sphere s2n-1. We choose two n-dimensional chains A and lJ in the ball D 2n , the boundaries of which coincide with the cycles a and b respectively and which lie wholly inside the ball D 2n , with the exception of their boundaries. In this case we can make sense of the The topology of the non-singular level set and the variation operator of a singularity 39 intersection number (A 0 B)D of the chains A and B in the ball D 2 n and /(a,b)=(Aob)s =( -1)n(A OB)D =(BoA)D =( -1)n/(b,a).

2. For Izi < 4 the level set Fz is transverse to the (2n -1 )-dimensional sphere Sz = (Jliz (the level set Fz is a manifold for Z =F 0, the zero level set Fo is a manifold everywhere except zero). Elements of the theory of Picard-Lefschetz 23 Proof. Let x E Fz n S2 and suppose that the level set Fz is not transverse to the sphere S2 at the point x. Then d,z(x) is linearly dependent on df(x) and df(x), that is d,z(x)=rxdf(x)+/3df(x), where rx, /3Eer. We have df(x) = 2~xjdxj, dJ(x)=2~xjdxj, d,z(x)=~xjdxj +~xjdxj, from which it follows that But not all coordinates Xj equal zero.

ED g near the critical point. 9. The homomorphismj. (f ED g) of the singularity! (g) can be explained in the following manner~ We consider the function! (f ED g) (more precisely we consider the function! 0 1t1, where * The topological structure of isolated critical points of functions 56 is the projection on the first factor)~ The preimage (jo1td- 1(z) of the pointzE ce consists of points (x, Y) E cen E9 cem, for which I(x)=z, g(y) =e-z. Therefore (if we ignore the details connected with the radii of the balls in which we consider the non-singular level manifolds of the functions) The mapping is non-degenerate outside the preimages of the points 0 and e.

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A conjecture in arithmetic theory of differential equations by Katz N.M.


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