When Nine Points Are Worth But Eight: Euler’s Resolution of Cramer’s Paradox - Higher Order Equations

Author(s):
Robert E. Bradley (Adelphi University) and Lee Stemkoski (Adelphi University)

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 a polynomial equation of the form $p(x,y)=0$ 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

$A x + B y + C = 0.$

The general form of a polynomial equation of degree two (a quadratic equation) was written by Euler as

$\alpha y^2 + \beta xy + \gamma x^2 + \delta y + \varepsilon x + \zeta = 0.$

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 $\alpha_{k,i}$ 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, the 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 $n.$  An equation of the form $p(x,y)=0$ 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, denoted $\varphi_n,$ is one less than the number of coefficients; that is $\varphi_n = \frac{n^2+3n}{2}.$

Robert E. Bradley (Adelphi University) and Lee Stemkoski (Adelphi University), "When Nine Points Are Worth But Eight: Euler’s Resolution of Cramer’s Paradox - Higher Order Equations," Convergence (February 2014)