arxiv: 2605.04910 · v1 · submitted 2026-05-06 · 🧮 math.RA · math.AC· math.OC

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Symmetric Bessmertnyu{i} Realizations and Field Extension Problems in Characteristic 2 - A Differential Algebra Approach

Aaron Welters, Soumya Sinha Babu

Pith reviewed 2026-05-08 16:02 UTC · model grok-4.3

classification 🧮 math.RA math.ACmath.OC

keywords Bessmertnyĭ realizationssymmetric realizationscharacteristic 2differential algebrafield of constantsSchur complementsfield extensionsrational functions

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The pith

Formal partial derivatives on multivariate rational functions reduce symmetric Bessmertnyĭ realization problems in characteristic 2 to scalar conditions on diagonal entries alone.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper gives a short algebraic proof that symmetric Bessmertnyĭ realizations exist in characteristic 2 by introducing formal partial derivatives on rational functions and studying the associated fields of constants. This machinery produces explicit scalar criteria that replace the original matrix-valued conditions, so that realizability is decided entirely by the diagonal entries of the associated symmetric matrix pencil. The same criteria also yield a new result on field extension problems for both symmetric and homogeneous symmetric cases. In the scalar setting the realizable functions are shown to form vector spaces over these constant fields, and the paper counts the many counterexamples that appear only in characteristic 2. Readers interested in linear systems or algebraic geometry over finite fields gain a direct computational shortcut that avoids algorithmic search.

Core claim

By defining formal partial derivatives on multivariate rational functions over fields of positive characteristic and taking the corresponding field of constants, one obtains scalar criteria for symmetric and homogeneous symmetric realizability in characteristic 2; these criteria reduce the matrix-valued realization problem to conditions on the diagonal entries alone.

What carries the argument

The field of constants of formal partial derivatives on multivariate rational functions over a field of characteristic 2, which extracts the necessary scalar conditions from the entries of a symmetric matrix pencil.

If this is right

The Symmetric Bessmertnyĭ Realization Theorem holds in characteristic 2.
A new theorem classifies the possible field extensions that admit symmetric or homogeneous symmetric Bessmertnyĭ realizations.
Realizable scalar rational functions are precisely the vector spaces over the appropriate fields of constants.
The proportion of non-realizable rational functions in the scalar case is explicitly quantified for characteristic 2.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same derivative-and-constant-field construction may supply scalar tests for realizability questions in other positive characteristics.
State-space models in linear systems theory over characteristic-2 fields become easier to verify once only diagonal rational functions need inspection.
The reduction to diagonals suggests that off-diagonal coupling in symmetric pencils is automatically satisfied once the scalar criteria hold.

Load-bearing premise

Formal partial derivatives on multivariate rational functions over positive-characteristic fields produce a field of constants that fully captures symmetric realizability without extra obstructions arising from characteristic 2.

What would settle it

A concrete symmetric matrix pencil over a field of characteristic 2 whose diagonal entries satisfy the constant-field conditions but whose Schur complement fails to be a symmetric Bessmertnyĭ realization, or vice versa.

read the original abstract

We present a short, purely algebraic proof of the Symmetric Bessmertny\u{i} Realization Theorem in the characteristic $2$ case recently proved in [EOW26]. Symmetric Bessmertny\u{i} realizations are Schur complements of affine linear symmetric matrix pencils, and they arise naturally as state-space representations in linear systems theory. In contrast with the algorithmic approach in [EOW26], we use differential algebra: by defining formal partial derivatives on multivariate rational functions over fields of positive characteristic and considering their corresponding field of constants, we obtain scalar criteria for symmetric and homogeneous symmetric realizability in characteristic $2$, effectively reducing the matrix-valued problem to its diagonal entries. As a consequence, we prove a new theorem on the field extension problem for symmetric and homogeneous symmetric Bessmertny\u{i} realizations. Finally, in the scalar case, we identify realizable rational functions with vector spaces over appropriate fields of constants and quantify the abundance of counterexamples in characteristic $2$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a differential-algebra proof of the symmetric Bessmertnyĭ realization theorem in char 2 plus a new field-extension result, but the sufficiency of the diagonal constant-field conditions needs explicit verification there.

read the letter

This paper supplies a differential-algebra proof of the symmetric Bessmertnyĭ realization theorem in characteristic 2 and a new theorem on the associated field-extension problem. The proof is short and algebraic. It defines formal partial derivatives on rational functions over fields of positive characteristic, takes the corresponding constant field, and reduces the symmetric realizability question to scalar conditions on the diagonal entries alone. This is a clear contrast to the algorithmic route in EOW26. The same method yields criteria for the homogeneous symmetric case and, in the scalar setting, identifies the realizable functions with vector spaces over the constant fields while giving a count of counterexamples in char 2. The soft spot is the characteristic-2 behavior of those derivatives. Since they annihilate all squares, the constant field properly contains the squares of the indeterminates. The claim is that requiring the diagonals to be constant still gives necessary and sufficient conditions for the existence of the full symmetric pencil. That step has to be verified directly; otherwise the scalar test could accept cases where the off-diagonal Schur-complement entries fail to cooperate. The abstract asserts there are no anomalies, so the manuscript presumably contains the check, but it is the part that needs the most scrutiny. The work is aimed at specialists in algebraic realization theory and positive-characteristic algebra. Anyone already following Bessmertnyĭ realizations or differential methods in systems theory will find the reduction and the new field-extension theorem useful. It is solid enough on its own terms to deserve a serious referee. I recommend sending it to peer review. The new proof technique and theorem are concrete, and the only real question is whether the char-2 reduction is fully rigorous.

Referee Report

1 major / 2 minor

Summary. The manuscript presents a short algebraic proof of the Symmetric Bessmertnyĭ Realization Theorem in characteristic 2 using differential algebra. Formal partial derivatives are defined on multivariate rational functions over fields of positive characteristic; the associated field of constants yields scalar criteria for symmetric and homogeneous symmetric realizability, reducing the matrix-valued problem to conditions on the diagonal entries. As a consequence, a new theorem on the field extension problem is proved, and in the scalar case realizable rational functions are identified with vector spaces over the constant fields, with the abundance of counterexamples in characteristic 2 quantified.

Significance. If the central reduction is valid, the work supplies a concise, purely algebraic alternative to the algorithmic proof in [EOW26] and establishes new results on field extensions for these realizations. The explicit identification of realizable functions with vector spaces over constant fields in the scalar case is a clear strength, as is the focus on characteristic-2 phenomena.

major comments (1)

[Proof of the Symmetric Bessmertnyĭ Realization Theorem in characteristic 2] The central reduction (that diagonal entries lying in the constant field of the formal partial derivatives are necessary and sufficient for symmetric Bessmertnyĭ realizability) is load-bearing. In characteristic 2 the formal partials annihilate p-th powers, so the constant field properly contains the subfield of squares. The manuscript must explicitly verify that this enlargement does not admit spurious solutions in which the off-diagonal entries of the Schur complement fail to satisfy the required algebraic relations; otherwise the scalar criteria are only necessary. This verification should appear in the proof of the main realization theorem.

minor comments (2)

The abstract states that the approach 'effectively reduc[es] the matrix-valued problem to its diagonal entries'; the precise statement of the scalar criteria (including any homogeneity assumptions) should be restated verbatim in the main text immediately after the definition of the constant field.
Notation for the field of constants in positive characteristic should be introduced with a short example contrasting the characteristic-2 case with characteristic 0 to aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying a point that requires greater explicitness in the central reduction. We address the major comment below and will revise the manuscript to incorporate the requested verification.

read point-by-point responses

Referee: [Proof of the Symmetric Bessmertnyĭ Realization Theorem in characteristic 2] The central reduction (that diagonal entries lying in the constant field of the formal partial derivatives are necessary and sufficient for symmetric Bessmertnyĭ realizability) is load-bearing. In characteristic 2 the formal partials annihilate p-th powers, so the constant field properly contains the subfield of squares. The manuscript must explicitly verify that this enlargement does not admit spurious solutions in which the off-diagonal entries of the Schur complement fail to satisfy the required algebraic relations; otherwise the scalar criteria are only necessary. This verification should appear in the proof of the main realization theorem.

Authors: We agree that an explicit verification is necessary to confirm sufficiency of the scalar criteria, given that the constant field in characteristic 2 properly contains the subfield of squares. In the proof of the main realization theorem, the necessity direction follows directly from the definition of the constant field. For sufficiency, the argument proceeds by constructing the symmetric matrix pencil whose Schur complement recovers the given rational function when the diagonal entries are constant; the off-diagonal entries are then determined algebraically and satisfy the required relations by the chain rule and the fact that the formal partial derivatives annihilate the relevant expressions. However, to address the referee's concern directly and rule out spurious solutions, we will insert a clarifying paragraph immediately after the statement of the theorem. This paragraph will explicitly compute the Schur complement entries in the presence of p-th powers and verify that they remain consistent with the algebraic relations demanded by Bessmertnyĭ realizability. The revised proof will therefore establish both necessity and sufficiency without relying on implicit steps. revision: yes

Circularity Check

0 steps flagged

No significant circularity in differential-algebra derivation

full rationale

The paper applies standard formal partial derivatives to multivariate rational functions over fields of positive characteristic and extracts scalar criteria from the resulting constant field. This construction is independent of the target realization conditions and does not reduce any claimed equivalence to a fitted parameter, self-definition, or prior self-citation. The reference to [EOW26] merely identifies the theorem being reproved; the new proof proceeds directly from the differential-algebra definitions without invoking uniqueness theorems or ansatzes from overlapping prior work. No load-bearing step collapses to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the newly introduced formal partial derivatives and the associated field of constants; these are domain assumptions rather than free parameters or invented entities.

axioms (1)

domain assumption Formal partial derivatives can be defined on multivariate rational functions over fields of positive characteristic such that the field of constants behaves as in characteristic zero.
Invoked to obtain scalar criteria by reducing to diagonal entries.

pith-pipeline@v0.9.0 · 5482 in / 1218 out tokens · 46717 ms · 2026-05-08T16:02:33.722181+00:00 · methodology

discussion (0)

Reference graph

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