I work on polynomial maps, computational algebra, and the verification of algebraic
claims by machine-checkable certificates. Recent work concerns the structure of the
counterexample to the Jacobian Conjecture announced in July 2026 — in particular degree
minimality within its equivariant class — and the certification of parameters of quantum
error-correcting codes.
The organising principle of this work is that a computational claim should travel with a
witness that a reader can check independently, without repeating the search and without
trusting the software that found it.
Preprints and notes
D. Kirtchakov, The minimum number of arc-disjoint transitive triples in
a tournament: a first certified determination of ν₃(9) = 9 and ν₃(10) = 12,
August 2026, 8 pp.
First certified determination of both values, confirming Yuster's 2004 formula at
n = 9, 10 by certified lower bounds over all tournament isomorphism classes with
replayable optimality certificates (one declared trust assumption: complete enumeration
by nauty's gentourng); upper-bound constructions due to Yuster (2004), credited.
Released August 5, 2026:
PDF ·
certificates ·
doi:10.5281/zenodo.21816010 · arXiv identifier to follow.
D. Kirtchakov, If a [[14,3,5]] stabilizer code exists, its monomial
automorphism group has order 2ᵃ3ᵇ5ᶜ, August 2026, 9 pp.
A certificate-backed exclusion of automorphism orders divisible by 7, 11, 13 for the
[[14,3]] distance-5 existence question, open since 2005 (codetables.de). The existence
question itself remains open and is not claimed; the CSS case is Koh et al.,
arXiv:2601.20927, credited.
Released August 5, 2026:
PDF ·
certificates ·
doi:10.5281/zenodo.21816018 · arXiv identifier to follow.
D. Kirtchakov, Replayable minimum-distance certificates for stabilizer
codes, August 2026, 17 pp.
Certified distances for eleven codes through IBM's bivariate-bicycle family at n = 288,
with no solver and no proof assistant in the trusted base. A verification contribution;
the codes and distance values are IBM's, credited.
Released August 5, 2026:
PDF ·
certificates ·
doi:10.5281/zenodo.21799780 · arXiv identifier to follow.
D. Kirtchakov, Degree minimality in the equivariant class of the
Alpöge Keller map, and the moment-map structure of its cotangent lift,
August 2026, 20 pp.
Three new results about the ℂ*-equivariant class of the degree-7 Keller map announced by
L. Alpöge in July 2026. First, a degree-minimality theorem: in that class no Keller map
of degree ≤ 6 exists at all in the sector containing Alpöge's map, and every degree-≤6
map elsewhere in the class is an automorphism — established by eight Gröbner-basis
unit-ideal certificates over ℚ, reproduced mod 32003, strengthening a partial result of
T. Shaska (arXiv:2607.20210, Thm 10.10), which settles the degree-one case and leaves the
general case open. Second, a no-go lemma valid at every degree and weight. Third, a
moment-map identity: the ℂ*-action lifts Hamiltonianly to the cotangent bundle and the
cotangent lift preserves the moment map exactly. The cotangent lift itself, its exact preservation of the
symplectic form, and the associated Weyl-algebra endomorphism are
W. G. P. Mayner's (July 2026), independently verified here; the note adds their
identification as a Poisson-conjecture witness and the quantization identity relating
them. The equivariant reduction, master equation, S₃ cover and image theorem were
obtained independently but are anticipated by T. Shaska (arXiv:2607.20210), by the
anonymous ulam.ai note, and by Mayner; priority is credited in the paper.
Released August 5, 2026:
PDF ·
certificates & scripts.
Archival doi:10.5281/zenodo.21799112 · arXiv identifier to follow.
Degree minimality across weight systems.
The equivariant reduction extends to the weight systems (1, −1, −k). Certificates for
k = 1 and k = 3 are in preparation.
Also under way (see the library for the current list): the
oriented Ramsey value k(3,4) ∈ {21, 22} for Erdős problem #112; a certified verification of
Kelmans' 1984 path-partition problem at n ≤ 20; and the [[14,3,5]] existence question inside
its remaining automorphism-free core.
Software and certificates
Every published claim is accompanied by the script that establishes it. The scripts assert
their results and fail loudly; the certificates are short enough to check by hand-written code
with no solver in the loop.
git clone https://github.com/05oz/certify
cd certify
python3 -m venv venv && venv/bin/pip install sympy
venv/bin/python scripts/core_verify.py # the map, its Jacobian, the collisions
venv/bin/python scripts/cover_verify.py # the cover and the image theorem
venv/bin/python scripts/weyl_verify.py # the Weyl-algebra endomorphism
Reproducing the Gröbner-basis certificates additionally requires
msolve. Instructions are in the
repository.
Correspondence
Corrections, prior references, and counterexamples are welcome and will be credited. If a
result here is already known, a pointer to the source is the most useful thing you can send.