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

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 \[\varphi_n = \frac{n^2+3n}{2}.\]