Polynomials in more than one variable

Let us begin by considering polynomials in two variables x and y

$$\begin{multline*} P(x,y) = p_{0,0} + p_{1,0} x + p_{0,1} y + p_{2,0} x^2 + p_{1... ...} y^2 + \dots + \\ + \dots + p_{n,0} x^n + \dots + p_{0,n} y^n \end{multline*}$$

where p_{k, l} are constants. The degree of a term p_{k, l}x^ky^l is defined as k + l if p_{k, l} is non-zero. The degree of a polynomial is the maximum of the degrees of each of its non-zero terms and the degree of the zero polynomial is undefined as before.

Exercise 22   Define the arithmetic operations on polynomials algorithmically so that polynomial manipulations can be implemented on a computer.

Polynomials can be ``evaluated'' to give functions; for any choice of a pair of constants (a, b), we can substitute x by a and y by b to obtain the ``value'' of the polynomial. By identifying the pair of constants (a, b) with the corresponding point in the plane, this gives us the function (on the plane) associated with the polynomial. The points where the function associated with the polynomial vanishes are called solutions of the polynomial.

In particular, a polynomial P(x, y) as above has the origin (0, 0) as a solution only if the constant term p_{0, 0} is zero. More generally, we say a polynomial vanishes to order m at the origin (0, 0) if all its terms have degree at least m.

The Binomial theorem allows us to extend this notion to points (a, b) other than the origin.

Exercise 23   Use the Binomial theorem to write a polynomial of degree n as a sum of terms of the form

$$\tilde{p}_{k,l}^{}$$(x - a)^k(y - b)^l

where k + l is at most n.

Thus we can say that a polynomial vanishes to order m at (a, b) if the terms in the above expression for it satisfy k + l $\geq$ m whenever the coefficients $\tilde{p}_{k,l}^{}$ are non-zero.

Exercise 24   If P and Q vanish to order m at a point, then so does R ^. P + Q for any polynomial R.

Because of this we can work with arithmetic operations on polynomials ``modulo'' terms that vanish to order m at a given point (a, b). In particular, any polynomial is like a linear polynomial upto terms that vanish to order two.

$$\begin{multline*} P(x,y) = \tilde{p}_{0,0} + \tilde{p}_{1,0} (x-a) + \tilde{p}... ...(x-b) + \\ \text{ terms that vanish to order 2 at $(a,b))$ } \end{multline*}$$

As before $\tilde{p}_{0,0}^{}$ is the value of P at (a, b).

Exercise 25   If we denote the rule P $\mapsto$ $\tilde{p}_{0,0}^{}$ by ($\partial$/$\partial$x)_{|(a, b)} then check that this is a constant derivation.

We also have polynomial derivations and as before

Exercise 26   Any polynomial derivation is determined by what it does to the two variables x and y.

In particular we have ($\partial$/$\partial$x) which is defined as the derivation that sends x to 1 and y to 0; $\partial$/$\partial$y is defined by symmetry.

Exercise 27   Repeat this subsection replacing two variables x and y with n variables (x₁,..., x_n).