By Fernando Q. Gouvea, Noriko Yui
The lawsuits of the 3rd convention of the Canadian quantity conception organization August 18-24, 1991
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Additional info for Advances in number theory: the proceedings of the Third Conference of the Canadian Number Theory Association, August 18-24, 1991, the Queen's University at Kingston
11. Let U be a connected open bounded set in en with piecewise /1, ... , fn be n functions holomorphic in some neighborhood U of U, without any common zeros on au. Then, the quantity C1 boundary. j (21l"i)n nu, () /\ d/1/\ ... /\ dfn, au where n denotes the differential form n I: (_1)k-1 Adf[k] nu,() = k=l IIf(()11 2n is a positive integer. FUrthermore, if /1 = . . = f n = 0 implies d/1/\ ... /\ dfn i=0, NU, U) is exactly the number of common zeroes of /1, ... , fn in the domain U. Proof. 18) to the function h = 1.
1\ d(jq. Th~re is a double sequence induced on the groups AP,q(X) by the operators 0 and 0, usually called the Dolbeault complex, and the corresponding cohomology groups are denoted HP,q(X). The homology groups Hp(X) can also be defined allowing the coefficients of the formal linear combinations to be real or complex numbers, instead of being integers. If one needs to make this fact explicit, one denotes them by Hp(X, Z), Hp(X, ~), Hp(X, e), respectively. Any element W E HP(X) defines a homomorphism Tw from Hp(X,e) into C by Tw(h):= 1 W where a is a representative of the homology class h.
In in U onto a neighborhood of the origin in en. For each such common zero a in U, there exists such a ball and such a diffeomorphism ()a. ,,---_ __ h() c:.. ~Jn(a)=o(Ja(8D(a,p(a)) where a(a)(w) = s(()~l(W)). This sum equals n(n-l) 2 (27ri)n(-1) (n - I)! h(a)= .. 9 when Zo = o. 11. Let U be a connected open bounded set in en with piecewise /1, ... , fn be n functions holomorphic in some neighborhood U of U, without any common zeros on au. Then, the quantity C1 boundary. j (21l"i)n nu, () /\ d/1/\ ...