Part 2 of 13. A linear change of variables scrambles the coefficients of a binary form, but its discriminant only picks up a power of the determinant. Noether's doctoral thesis studied invariants like this.
In 1907 Emmy Noether finished her doctoral dissertation at Erlangen under Paul Gordan, Über die Bildung des Formensystems der ternären biquadratischen Form, on the complete system of invariant forms of a quartic in three variables. This part works through the smallest cases of that subject: forms in two variables.
A binary quadratic form is . Its discriminant is . With the form keeps one sign and vanishes only at the origin; with it changes sign across two lines; with the two lines coincide.
Figure 1. The values of near the origin (y from −2 to 2): amber where positive, blue where negative. The lines are the real zeros of , solved from the coefficients. Set (for example a = 1, b = 2, c = 1) to watch the lines merge.
An invertible matrix acts on forms by substitution, . The new coefficients are messy polynomials in the entries of , but the discriminant only rescales:
An expression with this property is a relative invariant of weight w; on , where , it is invariant outright.
Figure 2. The substitution expanded with BigInt coefficients. Each row compares an invariant of the new form with times the invariant of the old one. Degree 4 has two basic invariants and as well as the discriminant, and the last line checks .
Plot how much a quantity grows against on log scales, and a relative invariant falls on a line whose slope is its weight. A plain coefficient does not.
Figure 3. Sixty seeded random real matrices with between about 0.06 and 16, applied to one fixed form. Each point is against ; lines are least-squares fits. Green: invariants. Red: the leading coefficient .
The slopes follow a rule: an invariant of degree in the coefficients of a form of degree has weight . For the quartic, have degrees 2, 3, 6. The ratio has weight 0, so it is unchanged by every invertible matrix, not just those of determinant one.
Over the complex numbers a binary form splits into linear factors, and the discriminant vanishes exactly when two of them coincide. Since , a real substitution also cannot change the sign of a cubic's discriminant, and that sign counts the real zero lines: three when positive, one when negative.
Figure 4. The family , optionally rotated. Top: the directions of its real zero lines for every , found by root-bracketing on the unit circle. Bottom: from the cubic formula, with its zeros marked. The disc draws the zero lines at the current . Rotating moves the lines but not the places where they merge.
The -orbit of a quadratic form lies on a level set of in coefficient space.
Figure 5. Four hundred seeded random matrices of determinant one, each applied to the starting form, plotted at the new coefficients . The dashed curve is , where ; since , every form with this discriminant lies on one side of it.
The discriminant does not quite name the orbit. and share , but a positive definite form stays positive under any real substitution, so the two orbits fill opposite quadrants and never meet.
Gordan proved in 1868, by explicit construction, that the invariants of any binary form are generated by a finite list; for the quartic that list is just and . In 1890 Hilbert proved finite generation for forms in any number of variables with an argument that produced no list at all, and Gordan distrusted it; Part 12 tells that story.
Hilbert's argument rests on an ascending chain condition for ideals in a polynomial ring. In 1921 Noether made that condition the axiom of a general theory of ideals, the subject of Part 11. The rest of Act I turns to physics, where the object a group acts on is a Lagrangian.