### The Geometry of the Discriminant of a Polynomial

R. W. D. Nickalls, R. H. Dye
1996 Mathematical Gazette
1 Our theme E VERYONE knows that the condition for the quadratic ax 2 + bx + c to have two equal roots, i.e. to have a repeated root, is that its discriminant ∂ 2 = b 2 − 4ac should be zero. We should remark, at the outset, that we are concerned only with ordinary polynomials whose coefficients are complex numbers. Indeed, little is lost if a reader assumes that all our polynomials are real, i.e. have real numbers for all their coefficients, though their complex roots must be considered as well
more » ... considered as well as their real ones. Though less at one's finger-tips nowadays, it has been known since the sixteenth century that a cubic ax 3 + 3bx 2 + 3cx + d has a repeated root, i.e. two or three equal roots, if and only if ∂ 3 is zero, where Cardan, del Ferro and Tartaglia found ∂ 3 when they discovered formulae for the three roots of the cubic in terms of square and cube roots of simple algebraic expressions of its coefficients. Their first step was to write y = ax + b and transform the original cubic into y 3 + 3Hy + G; this 'reduced' cubic has the same value for its ∂ 3 . Standard texts vary as to what they call the discriminant; some take ∂ 3 ; others prefer −∂ 3 or ∂ 3 /4. When Ferrari found expressions for the four roots of a quartic he obtained a corresponding ∂ 4 . Later, a determinantal formula was discovered for the algebraic