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2.3. The Derivative 69

(ii) as a ring of polynomials in the variable y over F[z].

Corresponding to these, we can consider two types of differentiation of a

polynomial f (x, y):

(i) partial differentiation with respect to x resulting in the partial deriva-

tive fi(x, y), with y treated as a constant.

(ii) partial differentiation with respect to y resulting in the partial deriva-

tive fY(x, y), with x treated as a constant.

For example, if f(z, y) = 5x3y2 + 32’~ - 2z3 + 4y - 7,

fs(x, y) = 15z2y2 + 6xy - 6x2

fy(x, y) = 102sy + 3x2 + 4.

Just as in the case of polynomials of one variable, we can consider deriva-

tives of higher order. For example, we can define

fk&,Y) = (f&(x, Y), f&Y Y) = (f&(x, Y>Y

with fyt and fgY defined similarly. What would these second order partial
derivatives be for the examples?
(a) Formulate and prove a conjecture concerning the relationship between
fsy and fyr.
(b) Define partial derivatives of the third order. How many distinct pos-
sibilities are there?
(c) Define partial derivatives of the kth order, for any positive integer k.
(d) Show that, for any polynomial of two variables, the kth order partial
derivatives all vanish for k sufficiently large. How is the minimum such
value of k related to the degree of the polynomial?
(e) We can formulate a version of Taylor’s Theorem in which f (x, y) can
be expanded about (a, b) in the form:

f (x, Y) = coo + clo(x - a) + COI(Y - b) + cu(x - a)’ + c12(x - Q)(Y - b)

+ c22(y - b)’ + 1...

Write down the form of the terms of higher degree and determine the
coefficients in terms of the partial derivatives of f (x, y) at (a, b).
(f) Generalize the results of this section for polynomials of more than
two variables.

E.24. Homogeneous Polynomials. In Section 1.5, we defined a poly-
nomial to be homogeneous of degree d if each of its terms had the same
degree d. For a polynomial of two variables, this is equivalent to requir-
ing that f(tx,ty) = tdf(x, y) ( see Exercise 1.5.2). Write down a number
of homogeneous polynomials of various degrees and compute for each the