MATH 202 B - Problem Set 4
نویسنده
چکیده
(4.1) Let P : R → R be a polynomial which does not vanish identically. Use Fubini’s Theorem to show, by induction on the dimension n, that Z(P ) = {x ∈ R|P (x) = 0} has Lebesgue measure equal to zero. proof By induction on n. For n = 1, consider a polynomial P ∈ R[X], and let d = degP . Then P has at most d roots, and Z(P ) = {x ∈ R|P (x) = 0} is finite, thus has measure 0. Now let n ≥ 1, and assume the result is true for n. Consider a polynomial P ∈ R[X1, . . . , Xn+1] and assume that P is not identically zero. Viewing P as a polynomial in R[Xn+1][X1, . . . , Xn] (i.e. a polynomial with coefficients in R[Xn+1]), we have that there exists A a finite subset of N, and a collection of polynomials Qα ∈ R[X1], α ∈ A such that P (X1, . . . , Xn+1) = ∑
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MATH 202 B - Problem Set 10
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