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Z n ) / | Z 1 | 2 4 - . . , 5 is the solid unit sphere with center 0 in R2n. 4) there is a point p in a neighborhood of 0 in Rn such that p is not the image of a critical point under M or M(2) and such tha t and d[Jf, s, 0] = d[Jf, 5, p] d[Jf (2 \ £ ( 2 \ 0] = d[M^\ £ ( 2 \ pi Regarding d[M,s,p] as the algebraic number of ^>-points, we may say: \d[M, s, p]\ is less than or equal to the number of points in [M~1(p)] n s. 3-2) it follows that the number of points in [M~1(p)] n s is less than or equal to d[Mi2\ £ ( 2 \ p] which equals d[M(2), S{2\ 6].

R' -f iy''. First let M be the mapping of R2 into R2 described by: z' = zk where k is a positive integer. Let S be a circle of center 0 (the origin) and arbitrary fixed radius r. To compute d[M'. & 0]. let Ne{0) bo a neighborhood of 0 such that if p e AT£(0), then d\ M, S. p] is defined and d\ M. S. p] = d[M, S, 0]. 3-1) zk = p has exactly k distinct solutions. /x, uy, vx. uy are zero. Thus by the Cauchy-Riemann equations, Jacobian J is positive. 2), the local degree is k. 44 TOPOLOGICAL TECHNIQUES IN EUCLIDEAN tt-SPACE Now let Mj be the mapping of R2n into R2n described by: z' = zki °j+l — ~; + l First d[M}, *ST, 0] is defined for all spheres S with center 0.

W h e r e / j , • • - . / n are real-valued functions and assume t h a t / x , • • - , / n can be extended to complex-valued functions, call these also j u • • - , / „ , of the COMPUTATION OF THE LOCAL DEGREE complex valuables z} = xi -f iyi (j — 1, • • •, n). z 49 Then 9tt is described by: z 'i = f i ( i > ' • - , z n ) K =/n(2i,' - ^ Z j . arJ/arf + ••• + z 2 g l ] , 5 = [ ( Z 1 , . . , Z n ) / | Z 1 | 2 4 - . . , 5 is the solid unit sphere with center 0 in R2n. 4) there is a point p in a neighborhood of 0 in Rn such that p is not the image of a critical point under M or M(2) and such tha t and d[Jf, s, 0] = d[Jf, 5, p] d[Jf (2 \ £ ( 2 \ 0] = d[M^\ £ ( 2 \ pi Regarding d[M,s,p] as the algebraic number of ^>-points, we may say: \d[M, s, p]\ is less than or equal to the number of points in [M~1(p)] n s.