An Effective Lojasiewicz Inequality for Real Polynomials
نویسنده
چکیده
Example 1. Set f1 = x d 1 and fi = xi−1 − x d i for i = 2, . . . , n. Then Φ(x) := maxi{|fi(x)|} > 0 for x 6= 0. Let p(t) = (t d , t n−2 , . . . , t). Then limt→0 ||p(t)||/|t| = 1 and Φ(p(t)) = t d . Thus the Lojasiewicz exponent is ≥ d. (In fact it equals d.) This works both over R and C. In the real case set F = ∑ f 2 i . Then degF = 2d, F has an isolated real zero at the origin and the Lojasiewicz exponent is 2d.
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