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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
() 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 ”.)
(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.
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