Page 31 - Textos de Matemática Vol. 44

NATA´LIA BEBIANO’S LEGACY 21
When reporting the previous paper, we saw that if ux + vy + w = 0 is the equation of a supporting line of one of the convex sets W+(A),W+ (A), then
J −J FAJ(u,v,w)=det(uHJ +vKJ +wIn)=0.
Now Γ∗ = {(u,v,w) ∈ PC2 : FAJ(u,v,w) = 0} determines a dual curve Γ={(x,y,z)∈PC2 :xu+yv+zw=0, isatangentofΓ∗}
whose affine image CJ (A) = {(x, y) ∈ R2 : (x, y, 1) ∈ Γ} is the associated curve of WJ (A). For the case J = ±In Kippenhahn proved in 1951 that CJ (A) generates WJ (A) as its convex hull. In this paper such results are presented for the case r ̸= 0, n.
Recall that in the homogeneous coordinates of projective geometry a circle has an equation x2 + y2 + axz + bxz + cz2 = 0. So every circle contains the complex points (1, −i, 0), (1, i, 0), which lie due to their third coordinate being equal to 0 ‘at infinity’ . This leads to calling them the circular points at infinity. By means of these points one defines the concept of a focus of an algebraic curve: a point P ̸= (1, −i, 0), (1, i, 0), is focus of such a curve if the lines through P and the circular points of infinity are tangent to the curve. This designation is justified since in case of curves of order two - conics - these points P can be shown to be the what we all know as foci.
In analogy to what Murnaghan and Kippenhahn had proved for the classical case, the authors find:
Proposition. The n real foci of the algebraic line-curve defined by the equation FAJ (u, v, w) = det(uHJ + vKJ + wIn) = 0 are the eigenvalues of the matrix A=HJ +iKJ.
Given matrices A, B of the same size m × n, the second mixed compound is the m × n matrix with entries naturally double indexed by the lexicograph-
22
ically ordered 2-subsets of {1, 2, . . . , m} and {1, 2, ..., n} via
1 air ais bir bis
C2(A, B){i,j},{r,s} = 2 bjr bjs + ajr ajs .
Using this concept, authors derive the following, using Fiedler’s ideas.
Theorem. Let A = HJ + iKJ as above, if the line-curve FAJ (u, v, w) = 0 is irreducible, then as corresponding point-curve it is given by the non-linear part of the equation
C2(HJ − xIn, HJ − xIn) C2(HJ − xIn, KJ − yIn)
JJ JJ =0. C2(H −xIn,K −yIn) C2(K −yIn,K −yIn)