Sharp completion bounds for polynomial values of matrices
September 9, 2026
Abstract
Let be an algebraically closed field of characteristic zero and a nonconstant polynomial. For each , take the smallest multiplicity of a root of , and let be the largest of these numbers. We prove that every prescription of rows of an matrix can be completed to a value if and only if . Equivalently, for every rank- matrix , the image of is the right ideal exactly under this inequality. We deduce necessary and sufficient conditions for surjectivity and image linearity of sums when the coefficient column spaces form a direct sum. For complementary coefficients and powers, these conditions are sharp in every dimension. The proof combines the chain decomposition of a partial linear operator with a dimension estimate for Jordan blocks under polynomial evaluation.
1. Introduction
Polynomial maps with matrix coefficients extend the usual matrix Waring problem. Panja, Saini and Singh [1] determined the images of on matrices over an algebraically closed field. Gangwal, Mandal and Verma [3, Question 1.1] ask for surjectivity and image linearity in arbitrary dimension over infinite division algebras, and treat real and quaternion matrices of size two. Saini and Singh [2] study the distinct regime in which one coefficient is invertible. For , they prove the necessary condition
and prove sufficiency in dimensions at most four. Their Question 6.1 asks whether sufficiency holds in all dimensions.
We study the regime in which the coefficient column spaces form a direct sum. In particular, both coefficients of a two-term map may be singular and have complementary column spaces. We obtain a complete classification in this regime, with arbitrary scalar polynomials in place of powers. The underlying result is a sharp uniform completion theorem. The decomposition of partial operators used in its proof is classical Kronecker theory; the contribution is the polynomial completion bound and its application to maps with coefficients.
Throughout, is algebraically closed of characteristic zero, , and is nonconstant. Define
The minimum exists because is algebraically closed. The maximum is finite, lies between and , and equals unless some fiber of has no simple point. For , one has .
Theorem 1.1.
Let have rank . Then
Equivalently, for an -dimensional subspace of an -dimensional space , every linear map is the restriction of for some if and only if this inequality holds.
The equivalence with prescribed rows follows by transposition. The theorem concerns a uniform assertion over every prescription. An individual prescription can have a completion even when the inequality fails.
Theorem 1.2.
Let , let be nonconstant, and suppose that the column spaces form a direct sum . Set and
Then the following are equivalent:
The image of is a vector subspace of .
The image of is .
For every , one has .
Consequently, is surjective on if and only if and these inequalities all hold.
Neither theorem assumes that a coefficient is an idempotent or that different coefficients commute. The direct-sum hypothesis is on their images as linear maps. It is essential to the proof of Theorem 1.2. We do not settle the case of overlapping column spaces or Question 6.1 of [2].
2. Polynomial evaluation on Jordan blocks
Write for a Jordan block of size and eigenvalue . A block of size zero is omitted from a direct sum.
Lemma 2.1.
Suppose and . The Jordan type of is the Jordan type of . In particular, for and ,
Every matrix whose eigenvalues satisfy is a value of on a matrix of the same size.
Proof.
Put . Taylor expansion gives
The matrix is invertible and commutes with . Thus, for every , the nullity of is the nullity of . The nullities of all powers determine a nilpotent Jordan type.
The chains of are the residue classes of the indices modulo . When , one chain has length and the other chains have length . This proves (3). When , polynomial evaluation preserves the size of the Jordan block. Choose a simple preimage of each eigenvalue, apply this observation to every block, and conjugate the resulting direct sum.
Lemma 2.2.
There is at most one for which .
Proof.
Let . If all roots of have multiplicity at least two, this polynomial has at most distinct roots. The total multiplicity of these roots as zeros of is therefore at least . Two different fibers are disjoint and would contribute at least zeros to , counted with multiplicity. This contradicts .
The proof of Theorem 1.1 needs only finiteness of the exceptional set. Lemma 2.2 also makes the parameter easy to interpret. For example, has , although its degree is five: its zero fiber has minimum multiplicity two, and no other fiber is exceptional.
3. Decomposing a partial operator
We give the special case of the Kronecker decomposition needed below, including an elementary proof. For the general matrix-pencil decomposition, see Verdier [4, Section 8].
Lemma 3.1.
Let be finite-dimensional vector spaces and linear. There is a basis of adapted to consisting of:
a basis of an -invariant space ;
open chains
where , and precisely the first vectors belong to the adapted basis of ;
additional vectors outside .
If , , , is the number of chains and the number of additional vectors, then
Proof.
Set , , and
These spaces form a descending chain. If , then is -invariant and all subsequent spaces equal it. Set . For , the map induces an injection
Indeed, , and for is equivalent to .
Starting at and proceeding downwards to , choose representatives for a basis of . At each stage retain the images under of the representatives already chosen one level higher, and extend them to a quotient basis. This is possible by (5). Every newly added representative at level starts an actual chain of successive -images with vectors in and one final vector outside . Representatives propagated from higher levels continue an existing chain.
At each level the vectors of these chains give a basis of . Together with a basis of , all chain vectors lying in therefore form a basis of . By the injection at , the final vectors of the chains are linearly independent modulo . Thus the basis remains independent after these final vectors are adjoined. Complete it to a basis of . All chain equations are exact, since the representatives were propagated by the actual map . The dimension formulas follow by counting basis vectors.
The case is immediate. Put . If , extend arbitrarily to an endomorphism of and apply Lemma 2.1 to obtain . The required inequality is automatic since .
Suppose and
Apply Lemma 3.1. On each open chain, choose the image of its final vector so that the completed block is a companion matrix with distinct eigenvalues avoiding the critical values of . Such a companion matrix exists: choose any required number of distinct elements outside this finite set and use their monic product as the characteristic polynomial. The completed block has an -preimage in the same dimension, by Lemma 2.1.
It remains to handle . Split into Jordan blocks. For a block , choose a preimage of with multiplicity . Lemma 2.1 shows that this block, together with additional dimensions, is a value of : put blocks of size and eigenvalue on those dimensions. The resulting direct sum is similar to . Transporting the preimage by this similarity gives the specified action of on the original block.
If no block requires additional dimensions, we are done. Otherwise . If is the total number of additional dimensions required, then
For this estimate, retain only blocks with and . Their total size is at most , and their number is at least one.
There are unused basis vectors after completing the open chains. Equations (4) and (6) give
We may therefore use these vectors for all the required Jordan blocks. On any remaining vectors use scalar blocks, which are values of . The direct sum of these constructions is an endomorphism extending .
For necessity, choose with and prescribe on . If extends , then is -invariant and the minimal polynomial of is divisible by . Hence has a -Jordan block of length at least . For each Jordan block of with , Lemma 2.1 bounds the largest resulting block by
Thus , which is equivalent to .
Passage to the coefficient formulation.
The space consists of the matrices with . For such a , put and define
This is well defined because . Every linear map is obtained this way. Moreover,
The necessity argument is the same Jordan-size obstruction that appears in [2, Proposition 4.2] for power maps with one invertible coefficient. The sufficient completion argument supplies the other direction for the one-sided image problem.
Write and . The direct-sum hypothesis gives a linear isomorphism
Under this isomorphism, is the product of the individual images . Equality with is therefore equivalent to equality with in every component. Theorem 1.1 proves the equivalence of assertions 2 and 3.
For assertion 1, let be the set of matrices whose eigenvalues avoid the finitely many critical values of . This is a nonempty Zariski open set, and Lemma 2.1 gives . Its image under the surjective linear map is Zariski dense in . For instance, any polynomial vanishing on that image pulls back to a polynomial vanishing on the dense set , and hence vanishes identically. Thus each individual image is dense in , and is dense in . A vector subspace of a finite-dimensional vector space is Zariski closed. Consequently assertion 1 implies assertion 2; the reverse implication is immediate. The surjectivity statement follows.
Corollary 5.1.
Suppose , with ranks and , and let . Then is surjective if and only if
Whenever either inequality fails, its image is Zariski dense in but is not a vector space. For , surjectivity is equivalent to .
Proof.
Apply Theorem 1.2 with , , and use . In the square case the two strict inequalities are equivalent to and .
For a small illustration, let , and . The map is surjective, whereas is not. Indeed, the latter misses : the first two rows of would have to be those of , so would have a nilpotent Jordan block of length at least two. No cube of a matrix can have such a block at zero. The same obstruction is already implicit in the general necessary condition of [2]; the example illustrates the classification rather than claiming priority for failure of the older surjectivity conjecture.
As a higher-degree example, replacing either square by leaves its threshold unchanged. More generally the degree of a scalar polynomial is replaced by the minimum multiplicity in its exceptional fiber, if that fiber exists.
6. Checks and remaining questions
The proof above is independent of computation. The accompanying exact-arithmetic script checks the partial-operator decomposition for all maps and for additional maps over . It classifies all matrices in by whether they have square or fourth roots over the algebraic closure, using nilpotent Jordan partitions. All prescriptions of two rows admit a square completion; admit a fourth-power completion, and the displayed nilpotent prescription is among the eight exceptions. The script also checks the dimension estimate and the balanced Jordan-block formula over explicit finite ranges. These checks are diagnostics for the universal proof, not substitutes for it.
Theorem 1.1 also gives a sharp criterion for the relaxed map , where the second variable enters linearly: it is surjective precisely when . Indeed, after quotienting the target by , the problem is exactly a prescription on the dual subspace annihilating . Requiring itself to be a -th power is an additional condition. Establishing that additional step would be necessary to resolve [2, Question 6.1]; our completion theorem does not establish it.
For general coefficient families with intersecting column spaces, a target no longer has unique components. Determining which decompositions admit polynomial preimages remains a natural next problem. Over non-algebraically-closed fields, arithmetic conditions on scalar preimages also enter, so the present rank criteria are not asserted there.
Research disclosure.
This manuscript was prepared with OpenAI Codex assistance in problem selection, proof development, literature search, computation and writing. It has not undergone independent expert review or journal peer review. The accompanying assessment separates the proved manuscript claims from unresolved questions and historical-priority limitations.
References
[1] S. Panja, P. Saini and A. Singh, Images of polynomial maps with constants, Mathematika 71 (2025), no. 3, e70031. Preprint: arXiv:2312.13865.
[2] P. Saini and A. Singh, Polynomial maps with constants on matrix algebra, preprint (2026), arXiv:2604.27592v1.
[3] A. Gangwal, A. Mandal and S. Verma, Polynomial maps with constants over division algebras and the generalized Kaplansky–L’vov conjecture, preprint (2026), arXiv:2607.27226v1.
[4] O. Verdier, Reduction and normal forms of matrix pencils, preprint (2012), arXiv:1205.1138, Section 8.