@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"

DeepMind's CEO told Prince Harry his AI lab now employs 700 staff: 'It’s really the biggest collection of brain power anywhere in the world on this topic' / businessinsider

blade runner 2049 minireview / Tmachine