Math 413/813 Homework Assignment 1 due date: Jan. 14, 2015 1. Suppose that X ⊂ Rn is a shape. (a) If f1 and f2 are functions on Rn , show that f1 = f2 on X (i.e., when restricted to X) if and only if f1 − f2 is zero on X. (b) If g is a function on Rn which is zero when restricted to X, and h any function on Rn , show that hg is zero when restricted to X. (c) Now let X be the circle {(x, y) ∣ x2 + y 2 = 1} ⊂ R2 . Take the following functions on R2 and organize them into groups according to their equality when restricted to X: () 1; () y; () x2 + y 2 ; () x2 − y 2 ; () 2x2 + 1; () 2x2 − 1; () x4 − y 4 ; () y 3 + x2 y. (I.e, group together the functions which are equal when restricted to X.) [Math 813 only] (d) Let X be the unit circle as in part (c). Let f (x, y) be any polynomial in x and y. Prove that there is a polynomial of the form g(x, y) = g0 (x) + g1 (x)y such that the restriction of f to X is equal to the restriction of g to X. 2. Let X be the unit circle {(x, y) ∣ x2 +y 2 = 1} ⊂ R2 and Y the unit sphere {(u, v, w) ∣ u2 + v 2 + w 2 = 1} ⊂ R3 . Define a map ϕ∶ X Ð→ Y by the rule ϕ(x, y) = (xy, y 2 , x). (a) Show that ϕ is well-defined. That is, show that if (x, y) ∈ X then ϕ(x, y) ∈ Y . (b) Compute ϕ∗ (u), ϕ∗ (v), and ϕ∗ (w). (c) Compute ϕ∗ (3u2 − 2vw + 5). (d) Let f be the function 5xy 3 +7x2 −9y 2 restricted to X. Find a polynomial g(u, v, w) on R3 so that f = ϕ∗ (g). 3. Let X = R and Y = R2 . The ring of polynomial functions on X is R[x]. The ring of polynomial functions on Y is R[x, y]. (a) The ring R[x] is a subring of R[x, y], i.e., the inclusion map ψ1 ∶ R[x] Ð→ R[x, y] is a ring homomorphism. Find a map ϕ1 ∶ Y Ð→ X such that pullback by ϕ1 induces ψ1 . (I.e., “ϕ∗1 = ψ1 ”.) 1 (b) The map ψ2 ∶ R[x, y] Ð→ R[x] given by “setting y = 0” (i.e., ψ2 (f (x, y) = f (x, 0)) is also a ring homomorphism. Find a map ϕ2 ∶ X Ð→ Y so that ϕ∗2 = ψ2 . (c) How would you describe these maps geometrically? (I.e., in a picture or in words, what do they do?) Minor suggestion: The fact that there is more than one x may make things more confusing. Relabelling one set of variables and describing the ring homomorphisms in the new variables may make things a bit clearer. 2

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