Explicit Noether Normalization for Simultaneous Conjugation via Polynomial Identity Testing

Mulmuley recently gave an explicit version of Noether’s Normalization lemma for ring of invariants of matrices under simultaneous conjugation, under the conjecture that there are deterministic black-box algorithms for polynomial identity testing (PIT). He argued that this gives evidence that constructing such algorithms for PIT is beyond current techniques. In this work, we show this is not the case. That is, we improve Mulmuley’s reduction and correspondingly weaken the conjecture regarding PIT needed to give explicit Noether Normalization. We then observe that the weaker conjecture has recently been nearly settled by the authors, who gave quasipolynomial size hitting sets for the class of read-once oblivious algebraic branching programs (ROABPs). This gives the desired explicit Noether Normalization unconditionally, up to quasipolynomial factors. As a consequence of our proof we give a deterministic parallel polynomial-time algorithm for deciding if two matrix tuples have intersecting orbit closures, under simultaneous conjugation. We also study the strength of conjectures that Mulmuley requires to obtain similar results as ours. We prove that his conjectures are stronger, in the sense that the computational model he needs PIT algorithms for is equivalent to the well-known algebraic branching program (ABP) model, which is provably stronger than the ROABP model. Finally, we consider the depth-3 diagonal circuit model as defined by Saxena, as PIT algorithms for this model also have implications in Mulmuley’s work. Previous work have given quasipolynomial size hitting sets for this model. In this work, we give a much simpler construction of such hitting sets, using techniques of Shpilka and Volkovich.

💡 Research Summary

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This paper revisits and substantially improves upon the explicit version of Noether’s Normalization Lemma for the ring of invariants of matrices under simultaneous conjugation, a problem originally studied by Mulmuley (2012). Mulmuley showed that constructing a small set of separating invariants – which would give an explicit normalization – can be reduced to a black‑box polynomial identity testing (PIT) problem for a certain algebraic circuit model. He further argued that derandomizing PIT for this model is likely beyond current techniques, coining the “GCT chasm”.

The authors first analyze Mulmuley’s reduction in detail and identify two places where it can be strengthened. The first improvement replaces the exponentially large generating set of invariants (all traces of products of the input matrices) by a much smaller set of separating invariants (T_{0}). While the full generating set guarantees separation, only a polynomial‑size subset is needed to distinguish any two orbits, provided the subset still generates a subring over which the whole invariant ring is integral. The second improvement observes that each invariant in (T_{0}) can be computed by a read‑once oblivious algebraic branching program (ROABP), a restricted form of algebraic branching program (ABP) where the variable order is fixed and each variable is read at most once.

The crucial technical breakthrough comes from recent work of the authors (Fiorini–Shpilka 2012) which constructs quasipolynomial‑size hitting sets for ROABPs. A hitting set is a deterministic set of evaluation points that guarantees a non‑zero polynomial computed by a small circuit will evaluate non‑zero on at least one point. Because such hitting sets now exist unconditionally, the authors can replace Mulmuley’s conjectural black‑box PIT algorithm with an explicit construction. Consequently, they obtain an unconditional explicit Noether Normalization for the simultaneous‑conjugation invariant ring, up to quasipolynomial factors. In concrete terms, they produce a polynomial‑size list of separating invariants with explicit algebraic circuits, and prove that the invariant ring is integral over the subring generated by this list.

Beyond the normalization itself, the paper derives an algorithmic application: deciding whether the orbit closures of two matrix tuples intersect. Using the separating invariants, one can evaluate a small set of polynomials on the two tuples and compare the results. This test can be performed in parallel poly‑time (NC(^2)), giving a deterministic polynomial‑time algorithm for the orbit‑closure intersection problem.

The authors also compare the strength of the PIT assumptions. They prove that Mulmuley’s original conjecture is equivalent to having deterministic black‑box PIT for general ABPs, a model strictly stronger than ROABPs. Since no subexponential hitting sets are known for ABPs, Mulmuley’s conjecture remains far stronger than necessary for the normalization task.

Finally, the paper revisits depth‑3 diagonal circuits (a class of depth‑3 arithmetic circuits where each multiplication gate computes a monomial raised to a power). Prior works gave quasipolynomial hitting sets for this model using sophisticated algebraic geometry. Here the authors present a much simpler construction, adapting the dimension‑reduction technique of Shpilka and Volkovich (2009). This yields explicit hitting sets of comparable size and further illustrates the versatility of the ROABP‑based approach.

In summary, the paper achieves three major contributions: (1) it weakens the derandomization hypothesis needed for explicit Noether Normalization from ABP‑level PIT to ROABP‑level PIT; (2) it leverages existing quasipolynomial hitting sets to give an unconditional, quasipolynomial‑time explicit normalization and a parallel algorithm for orbit‑closure intersection; and (3) it provides a streamlined hitting‑set construction for depth‑3 diagonal circuits. These results bridge a gap between algebraic geometry and computational complexity, showing that certain “hard” derandomization barriers can be bypassed with modern circuit‑complexity tools.