3.9 Show that k, the space of homogeneous polynomials of degree k in d, and k, the...

3.9 Show that k, the space of homogeneous polynomials of degree k in ℝd, and k, the space of all polynomials of degree ≤ k in ℝd have dimensions pH(k) = dim(k) = (k + d − 1 k

)

, pF(k) = dim(k) = (k + d k

)

, as stated in (3.16).

Hint: This exercise can be done using a simple counting argument (e.g.

Feller, 1968, p. 38). Write down a string of k + d − 1 characters, of which d − 1 characters are “|” and k characters are “x.” Let the number of

“x” characters between successive “|” characters indicate the power of successive components of t = (t[1], ..., t[d]). For example, with d = 4 and k = 5, one such string might be x|xx|xx|, corresponding to the monomial t[1]

1t[2]

2t[3]

2t[4]

0. Count the number of different arrangements of strings to get the dimension of k. For k, add an extra component t[d + 1] = 1 and note that a monomial of degree k in t[1], ..., t[d + 1] corresponds to a monomial of degree ≤ k in t[1], ..., t[d].