Consider four $n\times n$ matrices $a,b,c,d$ over finite field $F_q$ (or $F_p$ for simplicity), such that they satisfy three equations: $ac=ca,bd=db, ad-da=cb-bc $. Thus an affine algebraic manifold is defined ("Manin matrix variety").
Question 1: Is it true that number of such matrices over $F_q$ is polynomial in $q$ ? (I.e. the variety is "polynomial count" - which is quite rare phenomena for algebraic manifolds).
Question 1b: If yes - can nice generating function be produced similar to "commuting variety" ?
Question 2: Additionally we may ask if variety is: irreducible, normal and Cohen-Macaulay ? Similar to results/expectations about its "younger brother" - the "commuting variety" - just: $ac=ca$.
- For $2 \times 2$ matrices computations suggest the following polynomial: $ p^7 ( p^4+2p^3-p-1) $, checked for p=2,3,5,7,11,13,17,19.
- For $3 \times 3$ we only computed number for p=2: 6078976
Variations - similar to known results for commuting variety we can restrict blocks $a,b,c,d$ to be invertible, nilpotent, (anti)-symmetric, triangular, etc. and still expect the polynomial count property:
Invertibility. Additional constraint $a,b,c,d$ are invertible - polynom fit for 2x2 and $p=2,3,5,7$ - is: $p(p+1)(p−1)^4(p^5+p^4−3p^3+4p^2−6p+4)$ (verified by fast code of Peter Taylor based on Grobner basis. PS I had some doubts about it is polynomial - but resolved now - thanks to Peter).
Nilpotency. With additional constraint 2-niltpotency: $0=a^2=b^2=c^2=d^2$ : 2x2 case checked for (p=2,3,5,7,11,13,17,19) : $2p^5+p^4-2p^3$, 3x3 case checked for p=2,3,5: $4p^{10}+5p^9+p^8-6p^7-6p^6-p^5+2p^4+2p^3$ . For 3x3 case with just nilpotency: $2p^{13}+p^{12}+2p^{10}+3p^9+3p^8+p^6+p^5$
(ANTIsymmetric): 2x2 $p^4$ (obviously), 3x3 : $p^7+2p^6-p^4-p^3$ checked for p=2,3,5,7,11 .
Symmetric: 2x2 $2p^9+p^8-2p^7$ (checked p=2,3,5,7,11,13,17), 3x3: $4p^{15} + 2p^{14}− 3p^{13}−p^{12}− 2p^{11}+p^{9}$ checked for p=2,3,5.
Upper triangular with zero diagonal: 2x2 $p^4$ obviously, $2p^9+p^8−2p^7$ (same as 2x2 block upper-triangular matrices - probably due to standard isomorphism ) .
Uppertriangular: 2x2 $2p^9+p^8−2p^7$ (same as above for antisymmetric 3x3). Suspicious: $10p^{15}-4p^{14}+2p^{13}-2p^{12}-19p^{11}-p^{10}-p^9$ - coefficients are big, fit changes from p=2,3 to p=2,3,5, some other bad signs. So there are doubts that it is polynomial count variety.
Uppertriangular and squared to zero (a la A.A.Kirillov - see references below) 3x3 case - $2p^8+2p^7−2p^6−2p^5+p^4$ - checked p=2,3,5,7,11,13,17,19. (2x2 case: $p^4$ obviously).
Motivation: Manin matrices are beautiful object which satisfies many interesting properties and have various applications in representation theory, knot theory, etc, so general (exaggerated) expectation - whatever reasonable property we propose it will be true for Manin matrices (see example of Koszulity here (thanks to Steven Sam). In particular - it is a natural generalization of the "commuting variety" - so motivation for the questions - whether the properties known/expected for the "commuting variety" generalize to "Manin matrix variety" ?
Thanks/Related MO/MSE questions/papers:
The polynomials above were computed with the help of Peter Taylor code appeared in the mathoverflow answer on related question - code is very fast and efficient - allowing to guess polynom just by values at few points. There were other ways proposed - thanks to all (Peter Taylor, Roland Bacher, Max Alekseyev) who shared ideas !
Here Will Sawin, Ycor, David E Speyer, Anton Mellit shared approaches to tackle more simple case $AB+BA=0$, it might be some ideas in particular from Sawin's answer (which is the most general) might be used for the present question.
Here David E Speyer, Olivier Schiffmann provided very insightful answers on related more simple question - counting nilpotent commuting matrices and references to "Motivic Donaldson-Thomas invariants and McKay correspondence", "On the number of points of the Lusztig nilpotent variety over a finite field" context of these questions.
The question on generating function might be related to the so-called "exponential formulas" which is beautifully exposed in Qiaochu Yuan blog, see also q-analogue (thanks to Ofir Gorodetsky), $F_1$-analogue (thanks to Gjergji Zaimi), $e^{cycle} = permutation$.
Some remarkable materials on $F_q$ counting problems: Steven Sam "Counting matrices over finite fields" https://mathweb.ucsd.edu/~ssam/talks/2011/finitefield.pdf, Yifeng Huang "Generating function from counting mutually annihilating matrices and alike" https://yifeng-huang-math.github.io/files/talk_gocc.pdf
In 90-ies A.A.Kirillov was interested (1, 2,3) in count of unitriangular matrices $X^2=0$ motivated by the extension of his famous orbit method in representation theory to finite nilpotent groups of unitriangular matrices. He found connections of these counting polynomials to q-Hermite polynomials, Euler-Bernoulli numbers etc. Some his question was resolved by D.Zeilberger.
Various activity about "polynomial count variety" questions was motivated in part by M.Kontsevich conjecture disproved in "Matroids, motives and conjecture of Kontsevich".
Some more references here. (If you know more - please add).
Generalizations: The question above appeals to $2\times 2$ Manin matrices, it is natural to expect similar polynomial count phenomena for $n\times m$ Manin matrices (but not normality - because normality break for n-tuples of commuting matrices). One can also consider super-and-q-analogues of Manin matrices and cases $M(u)$-polynomial Manin matrices (which produce relations like here). As well as some generalizations to some other quadratics algebras (and their B,C,D versions due to A.Molev).
PS
The questions above arise quite years ago in our paper, but I am not aware of any progress.
