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1 (x) = F x = lim F (U) x∈U The space F x is called the stalk of F at x. For an element f ∈ F (U) its image fx ∈F x is called the germ of f in x. The topology in F is defined as follows: for each f ∈ F (U) the set {fx , x ∈ U} ⊂F is open. An arbitrary open set is a union of sets of this form. Properties: 1. The mapping φ is locally a homeomorphism of topological spaces. This is a characteristic property of sheaves. Indeed, let φ :F → X be a continuous mapping that is locally a homeomorphism. e.

We call a nbd V of a = (a1 , a ) suitable if V = D × B, where D = {|z1 − a1 | ≤ r} and B is a closed connected nbd of the point a and f is defined in V. The number r and B are so small that (i) f (z1 , z ) = 0 for z ∈ B and z1 ∈ ∂D and (ii) f (z1 , a ) has only one root z1 = a1 in D. Let m be the multiplicity of this root. We study the analytic set Z = {f (z) = 0, z ∈ V} 6 By Weierstrass Lemma we have f = φP where φ is holomorphic and does not vanish in V and P is a distinguished pseudopolynomial (pp) of order m at a for z ∈ B which means that P (z1 , z ) = (z1 − a1 )m + Am−1 (z ) (z1 − a1 )m−1 + ...

Am . S (p, q) = b0 ... bn−1 bn 0 ... 0 0 b0 ... b b ... 0 n−1 n ... ... ... ... 0 0 ... 0 b0 ... bn−1 bn 3 is called Sylvester resultant of the polynomials: R (p, q) = det S (p, q) . , bn with integer coefficients. Proposition 7 If A is a field, then R (p, q) = 0 if and only if there exist polynomials u, v ∈ A [t] such that uv = 0 and up + vq = 0 deg u < n, deg v < m. Proposition 8 If the field A is algebraically closed, then R (p, q) = 0 if and only if there exists, at least, one common root of p and q in A or at infinity.