@JarekDuda I finally found time to read that separate paper. I do like it. The only part which confused me a bit for the moment is section "D. Relative rotation of multiple polynomials". I hope it will become clearer after reading reference [4], i.e. your updated writeup on using polynomial invariants for graph isomorphism testing.
This is no evil joke, the first sentence which confuses me in that section is: "For example having two linear spaces A and B of symmetric matrices defining elipsoids as {x : x^T A x = 1}, we can use Frobenius inner product to translate geometry between them [4]." and the following explanation "Now for example performing Gram-Schmidt orthonormalization using such scalar product, if A and B differ only by rotation, this basis would be orthonormal for both of them." is not detailed enough for me.
@ThomasKlimpel, thanks, it is described more clearly in two weeks ago version of: arxiv.org/pdf/1703.04456
So we have these two subspaces \mathcal{P} = {a_1 P_1 + ... + a_d P_d} and analogously for Q, each describing set of ellipsoids to intersect e.g. {x: x^T P x = 1 for all P in mathcal{P}}
Choosing these P_i basis using Gram-Schmidt orthorormalization with such Frobenius inner product, we get Tr(P) = \sum_i a_i^2 "sphere normalization" and analogously for Q
After this sphere normalization, there has remained to check only rotation - Tr(P^k) and Tr(Q^k) are degree k homogeneous polynomials, testing if they differ by rotation can be made with the invariants ...
But the main question is still: how large d has to be to uniquely determine the set?
correction: there should be Tr(P^2) = \sum_i a_i^2 "sphere normalization"
@vzn I'm guessing you've gathered, but rules against spam apply in chat too, so please don't repost messages across multiple rooms unless you're really really sure people there are interested (especially if it's your own blog).
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