By David H. Sattinger

A dialogue of advancements within the box of bifurcation concept, with emphasis on symmetry breaking and its interrelationship with singularity concept. The notions of common recommendations, symmetry breaking, and unfolding of singularities are mentioned intimately. The booklet not just reports fresh mathematical advancements but in addition presents a stimulus for additional examine within the box.

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Rn. For example, a Hilbert basis for the symmetric polynomials in xt • • • xn is the set of elementary symmetric polynomials A Hilbert basis for the ring r of polynomials on the real line which are invariant under the reflection x —» -x isCT(X)= x 2 : every even polynomial can be written in the form h(x) = f(x2). As another example, consider the adjoint action of GL(n) on the vector space of n x n matrices given by A Hilbert basis for this action is the set of invariants 47 48 CHAPTER 3 These invariants are realized as the coefficients in the characteristic polynomials of A: Equivalently, one may take the invariants o- k (A)=TrA k , fc = l , .

We denote points in U" by x and the action of <$ by yx. A smooth mapping F:[R"—»R n is said to be equivariant if yF(x) = F(yx). A function h:R n -»IR is said to be invariant if h(yx) = h(x) for all xR " and ye<&. The space of smooth invariant functions h :R" —»1R forms a commutative ring with identity (h = 1) which we denote by I. Thus, if h, g e I then so do h + g and hg. The space of smooth equivariant mappings F: R" —» Rn forms a module over / which we denote by . A Hilbert basis for a ring of invariant polynomials is a set of invariant polynomials al,.

Points in S2""1 may be identified with points £ = ( £ l 5 . . , £n) e C" where Z"=i l£j| 2= 1- The group action is free, that is, it has no fixed points; the same was true of the action u —> — u on unit spheres. 14. Let j, k be integers, l^j'