Higher Order Equations

Now that we understand the relationship between a second order equation and the various kinds of conic sections, let's turn our attention to equations of higher order. A polynomial equation in two variables is an equation of the form $$p(x,y)=0,$$ where $$p(x,y)$$ is a polynomial. The terms of $$p(x,y)$$ have the form $$c x^i y^j$$, where $$c$$ is a constant coefficient and $$i$$ and $$j$$ are non-negative integers. We assume that $$p(x,y)$$ has been simplified so that there is only one such term for any particular pair $$i$$ and $$j$$. The degree of the term $$c x^i y^j$$ is $$i+j$$. Clearly, there can only be one term of degree 0, two terms of degree 1 and, in general, $$n+1$$ terms of degree $$n$$.

The degree of $$p(x,y)$$ is the maximum of the degrees of the terms of $$p(x,y)$$. Therefore, the general form of a polynomial equation of degree one (a linear equation) is $A x + B y + C = 0$ and the general form of the polynomial equation of degree two, as written by Euler as discussed previously, is $\alpha y^2 + \beta xy + \gamma x^2 + \delta y + \epsilon x + \zeta = 0.$ The general form of a polynomial equation of degree $$n$$ is $\sum^{n}_{k=0} \sum^{k}_{i=0} \alpha_{k,i} x^i y^{k-i} = 0,$ where the $$\alpha_{i,j}$$ are constant coefficients and there is at least one $$i_0$$ with $$0 \le i_0 \le n$$ satisfying $$\alpha_{n,i_0} \ne 0$$. Using the familiar formula for the sum of the first $$N$$ integers, number of coefficients in a polynomial equation of degree $$n$$ is the sum $1 + 2 + 3 + \ldots + n + (n+1) = \frac{(n+1)(n+2)}{2} = \frac{n^2+3n}{2} + 1.$

A curve of degree $$n$$ (called a "line" of degree $$n$$ by Euler and most other 18th century authors) is the graph of the solution set of a polynomial equation of degree $$n$$. A polynomial equation $$p(x,y) = 0$$ may be multiplied by an arbitrary non-zero constant without changing the set of pairs $$(x,y)$$ that satisfy it. Therefore, in order to determine its solution set, it's only necessary to specify the ratios among the coefficients of $$p(x,y)$$, not the coefficients themselves. The number of such ratios is one less than the number of coefficients, that is $\varphi_n = \frac{n^2+3n}{2}.$