Algebraic conditions and the sparsity of spectrally arbitrary patterns

: Given a square matrix A , replacing each of its nonzero entries with the symbol ∗ gives its zero-nonzero pattern . Such a pattern is said to be spectrally arbitrary when it carries essentially no information about the eigenvalues of A . A longstanding open question concerns the smallest possible number of nonzero entries in an n × n spectrally arbitrary pattern. The Generalized 2 n Conjecture states that, for a pattern that meets an appropriate irreducibility condition, this number is 2 n . An example of Shitov shows that this irreducibility is essential; following his technique, we construct a smaller such example. We then develop an appropriate algebraic condition and apply it computationally to show that, for n ≤ 7, the conjecture does hold for R , and that there are essentially only two possible counterexamples over C . Examining these two patterns, we highlight the problem of determining whether or not either is in fact spectrally arbitrary over C . A general method for making this determination for a pattern remains a major goal; we introduce an algebraic tool that may be helpful.


Introduction
A problem of active interest in combinatorial matrix theory is to relate combinatorial properties of a matrix to properties of the linear operator it represents. The combinatorial properties in question are often formulated in terms of some description of the matrix that depends only the signs or on the zero-nonzero character of its entries. A simple example is the zero-nonzero pattern of the matrix, which simply speci es which entries of the matrix are nonzero. This information can itself be represented as a matrix, de ned formally as follows.
The method used in [3] to prove that the aforementioned examples were spectrally arbitrary has become known as the Nilpotent-Jacobian Method. It was shown in [2] that this method cannot successfully be applied to any pattern with fewer than n nonzero entries.
When considering the situation for zero-nonzero patterns, it is useful to observe that the set of matrices over R with a given sign pattern is contained in the set of matrices that have the corresponding zero-nonzero pattern. It follows that a sign pattern cannot be spectrally arbitrary if its corresponding zero-nonzero pattern is not spectrally arbitrary. Hence, while Conjecture 1.7 asserts that no sign pattern with fewer than n nonzero entries is spectrally arbitrary, a stronger statement would be this same assertion for zero-nonzero patterns.
And the question of whether a particular zero-nonzero pattern is spectrally arbitrary may be considered over any eld. Thus, the following could be considered the ultimate strengthening of Conjecture 1.7. Conjecture 1.8 (The Generalized n Conjecture). Let F be any eld. Every irreducible n × n zero-nonzero pattern that is spectrally arbitrary over F has at least n nonzero entries. Conjecture 1.8 above, unlike Conjecture 1.7, includes the additional condition that the pattern is irreducible. This is necessary in the more general case where an arbitrary eld is considered; Shitov demonstrated this in [15] by showing that a certain reducible n × n zero-nonzero pattern with n − nonzero entries is spectrally arbitrary over C. (In Section 3, we revisit the argument given there and present a smaller such example.) Moreover, a theorem of [16] shows that the lower bound of n − still holds over every eld for zerononzero patterns that are irreducible, and the irreducibility is essential to the proof. For reference, we state this result below. Theorem 1.9 ([16,Theorem 3]). Let F be any eld. Every irreducible n×n zero-nonzero pattern that is spectrally arbitrary over F has at least n − nonzero entries.
Hence, to resolve Conjecture 1.8 it su ces to determine whether or not there exists an n × n irreducible zerononzero pattern with exactly n − nonzero entries that is spectrally arbitrary over some eld. A counting argument is enough to establish that this cannot occur over any nite eld, giving the following result. Theorem 1. 10 ([14]). Let F be a nite eld. Every irreducible n × n zero-nonzero pattern that is spectrally arbitrary over F has at least n nonzero entries.
As noted in [16], the existence of irreducible polynomials of arbitrary degree over every nite eld implies that the term "irreducible" can be removed from Theorem 1.10, since a reducible pattern cannot realize any irreducible polynomial, and therefore is already ruled out by the hypothesis that the pattern is spectrally arbitrary.
Returning to the case of R, we brie y survey the current state of knowledge. For all spectrally arbitrary patterns, the lower bound of n nonzero entries was proved in [4] for ≤ n ≤ , and extended in [5] to n = . Because this work proved the bound for both reducible and irreducible zero-nonzero patterns, it established both Conjecture 1.7 and Conjecture 1.8 over R for ≤ n ≤ . In addition, [5] established the bound for reducible zero-nonzero patterns (and hence also for reducible sign patterns) for n = and n = . For these values of n, we show that Conjecture 1.8 does hold, which also implies that Conjecture 1.7 holds in the irreducible case; since that was the only remaining case for n = and n = , our work here completes the proof of both conjectures for those values of n. Hence, what we prove here is su cient to establish that Conjecture 1.7 holds, and that Conjecture 1.8 holds over R, for ≤ n ≤ .
In fact, the results of the present paper subsume the case of R, as our work applies over all elds with either characteristic or characteristic p, except for some nitely many nonzero values of p. Conjecture 1.8 does hold over all such elds when ≤ n ≤ , and in the case of n = there are (up to some natural notions of equivalence) only two patterns that could be counterexamples. We examine these patterns in Section 7, where we show that they are in fact not spectrally arbitrary over R. Notably, however, the conjecture remains open over C for n = ; essentially the only patterns of this size that could be counterexamples are the two mentioned above. Determining whether or not either of these is spectrally arbitrary over C seems to be a di cult problem. In Section 5 we highlight an algebraic tool that may be useful.
Our con rmation of the n Conjecture over R for ≤ n ≤ is achieved via a computer-assisted veri cation outlined in Section 6. This veri cation exploits an algebraic condition, introduced in Section 4, that can be checked easily by most computer algebra systems to rule out the possibility that a particular zero-nonzero pattern is spectrally arbitrary. Beyond enabling computational results, such algebraic conditions may also be useful for proving general results about spectrally arbitrary patterns. In Section 5 we prove a lemma that may be useful in formulating further such conditions; in Section 7 we use this lemma to show that a particular pattern not ruled out by the computation is nevertheless not spectrally arbitrary over R.

Preliminaries
Let A be an n × n zero-nonzero pattern with m nonzero entries. Let A be the unique matrix with zero-nonzero pattern A in which each nonzero entry is one of the algebraic indeterminates x , x , . . . , xm, with the indices of these indeterminates occurring in increasing order with respect to the usual row-major ordering of the matrix entries. Letting p(z) be the characteristic polynomial of A, we have where each α i is a polynomial in the variables x , x , . . . , xm. Each α i is homogeneous, as each of its terms is the product of i distinct variables. Moreover, its coe cients are from the set { , − }, so that we may, for any eld F desired, consider this family of polynomials as belonging to F[x , x , . . . , xm]. When considering a pattern A, we refer to these as its associated polynomials.
That is, A is the matrix of indeterminates associated with the pattern A in the manner described above. Hence, calculation of the coe cients of the characteristic polynomial of A shows that for this pattern the associated polynomials are: A cursory glance at the polynomials above may reveal little reason to suspect that A cannot be spectrally arbitrary. But a deeper investigation reveals that In particular, this shows that α and α cannot be simultaneously zero while every x i is nonzero. Hence, no realization of A can have α = α = , and so A is not spectrally arbitrary.
In the above example, the pattern A is seen not to be spectrally arbitrary because it is not possible for all of its associated polynomials to vanish simultaneously without some x i being zero. That is, it fails the following necessary (but not su cient) condition for a pattern to be spectrally arbitrary.

De nition 2.2.
Let F be a eld. An n × n zero-nonzero pattern A is said to be potentially nilpotent over F if A has some realization A with entries in F and the characteristic polynomial x n .
We make use of the following standard connection between a square zero-nonzero pattern and a directed graph.

De nition 2.3. Let
A be an n × n zero-nonzero pattern. The digraph of A is the directed graph on vertices , , . . . , n in which there is an arc from vertex i to vertex j precisely when the (i, j)-entry of A is * . (Note that this digraph may have loops.) A realization of a zero-nonzero pattern A can be thought of as a way of assigning a nonzero weight to each edge of the digraph of A; the weight assigned to each arc is the value of the corresponding nonzero entry in the realization. One case of particular importance is where the nonzero entries are distinct algebraic indeterminates. The next example illustrates how these become weights for the arcs of the digraph. Figure 1 shows the digraph of the pattern A of Example 2.1 with weights given as distinct variables; this corresponds to the realization A given in (2).

Example 2.4.
The condition of a pattern being irreducible (see De nition 1.6) has a simple interpretation in terms of its digraph.

Theorem 2.5 ([11, Theorem 6.2.24]). A pattern is irreducible if and only if its digraph is strongly connected.
When considering a realization of an irreducible zero-nonzero pattern, the following theorem allows us to apply a helpful normalization. By the above theorem, when considering the question of which spectra are realized by some particular irreducible pattern, there exist n − entries which we may assume without loss of generality are equal to in an arbitrary realization of the pattern. This can be of tremendous advantage to the e ciency of computations. To illustrate such a normalization, we revisit the pattern from Example 2.1.

Example 2.7.
For the pattern A of Example 2.1, the arcs corresponding to x , x , x , x , x , x form a spanning tree as required by Theorem 2.6. Thus, we can normalize these entries in the realization A from (2.1) so as to give To see the e ect on the associated polynomials (3), we show these polynomials below, with each normalized variable replaced with a . (We display the s to highlight precisely where the normalization has an e ect.) Comparing the expressions for α and α above, we see that, once we neglect the term of −x x x that appears in the expression for α , the two di er by a factor of x . In particular, then, This example illustrates how the normalization made the relevant algebraic relationship between α and α easier to detect as compared with (3). What we gain from this is the knowledge that α and α cannot be simultaneously zero while all of the x i are nonzero. This implies that A is not potentially nilpotent, as no realization of A can have α = α = . Hence, A is not spectrally arbitrary.
The associated polynomials of A are determined (up to permutation of the x i ) by the digraph of A. This observation allows the computer search outlined in Section 6 to consider what happens algebraically for all patterns by conducting a systematic search of the strongly connected digraphs with n − edges.

Reducibility and the generalized n Conjecture
As mentioned in Section 1, the irreducibility of the pattern is a necessary condition in Conjecture 1.8. This was rst demonstrated in [15], by the construction of a reducible n × n zero-nonzero pattern that is spectrally arbitrary over C despite having only n − nonzero entries, for n = .
Here we show that the same idea can be used to construct such an example with n = .
The following fact is used explicitly in [15] and is a key to the argument there. Although the proof is basic, we include it here for the sake of completeness. Proof. The proof is by induction on n. For n = , the conclusion follows since p is univariate of degree k, and hence cannot have more than k roots in F. For n > , consider p as a polynomial in xn with coe cients in where t is the degree of p as a polynomial in xn (i.e., t is the highest power to which xn appears in any term of p). In particular, the total degree of q t is at most k − t. Hence, by the inductive hypothesis, any subset of S of size (n − ) + (k − t) = n + k − t − contains some n − values that can be assigned to the variables x , . . . , x n− so that q t does not vanish. In particular, when these values are taken for x , . . . , x n− in (4), the resulting polynomial in xn has degree t, and so there are at most t values on which this polynomial vanishes. But, setting aside the n + k − t − values from S that were already used, there are t + values remaining. Hence, one of these can be assigned to xn to complete the assignment of values to x , . . . , xn desired.
We now follow the argument from [15] to give a smaller example showing the necessity of the irreducibility condition in Conjecture 1.8.

Theorem 3.2.
For every n ≥ , there exists an n × n reducible zero-nonzero pattern with n − nonzero entries that is spectrally arbitrary over C. Moreover, when n = + k for some k ≥ , there is an n × n reducible zero-nonzero pattern with n − k nonzero entries that is spectrally arbitrary over C.

Proof. We begin with the × pattern
as its suitably-normalized generic realization. Then taking x = x = − and x = x = gives a nilpotent realization, while x = − i, x = + i, x = − , and x = − give a realization with characteristic polynomial (z− ) . Note that this nilpotent realization, together with scalar multiples of the latter realization, show that A realizes any characteristic polynomial with four identical roots.
Meanwhile, one may verify that A has the characteristic polynomial z + t z + t z + t z + t when its entries are chosen according to the equations Hence, a realization of A with this characteristic polynomial exists when all of the denominators in these equations are nonzero and the values for x through x given by the equations are nonzero. Those two conditions will be met provided since, in particular, the above condition ensures that the constant term of the quadratic (5) is nonzero, so that neither nor −t can be among its roots, ensuring that it is possible to choose a nonzero value of x according to (5) that still results in a nonzero value for x . We now proceed as in [15]. First, let λ , . . . , λ be the eigenvalues of A. Then each t i is given by (− ) i times the ith elementary symmetric polynomial in λ , . . . , λ . Hence, we may consider (6) as a polynomial in λ , . . . , λ . As in [15], we call this polynomial ψ. We see from (6) that the total degree of ψ (in terms of the variables λ j ) is . Therefore, by Lemma 3.1, given any distinct complex numbers, some subset of of them can be assigned as values to the λ j in such a way that ψ will not vanish, so that these give a spectrum realizable by A.
Therefore, if a set of complex numbers does not contain any subset of values giving a realizable spectrum of A, then it can contain at most distinct values, and (by the initial paragraph of the proof) each of those can occur at most times. Hence, such a set can have size at most . Thus, any set of at least complex numbers contains a spectrum realizable by A.
It follows from the above that the direct sum of A with any spectrally arbitrary pattern of order at least yields a (reducible) spectrally arbitrary pattern. In particular, results of [6,8] show that a particular wellstudied k × k zero-nonzero pattern denoted by T k is spectrally arbitrary for every k ≥ . Thus, for each n ≥ we have that A ⊕ T n− is a reducible zero-nonzero pattern of order n with n − nonzero entries that is spectrally arbitrary over C.
Building on the above, the direct sum of A with the spectrally arbitrary pattern A⊕T yields a spectrally arbitrary pattern of order n = with n − nonzero entries. Repeating this argument shows that the direct sum of T with k copies of A gives an n × n zero-nonzero pattern that is spectrally arbitrary over C and has n − k nonzero entries, with n = + k.

Algebraic conditions
In order for a pattern A to be spectrally arbitrary, it must allow a zero trace, which implies that the associated polynomial α cannot be a monomial. Of course, more generally, in order for A to be potentially nilpotent, none of the polynomials α i can be a monomial. This condition can be generalized even further; not only may none of the polynomials be a monomial, but in fact there cannot be any monomial in the ideal they generate.
That is, we have the following result of [18].
then A is not potentially nilpotent over F.
The notation used in (7) serves to emphasize that the ideal α , . . . , αn is considered as generated over F. In the case where F = Q, we can check condition (7) e ciently using any standard computer algebra package. The next result shows that this check over Q alone is su cient to rule out A being potentially nilpotent over all but a very limited class of elds.

Theorem 4.2. Let
A be an n × n zero-nonzero pattern with m nonzero entries and associated polynomials α , . . . , αn. If then A is not potentially nilpotent, except possibly over some elds of characteristic q ≠ , for nitely many values of q.
We rst consider the case where F is a eld of characteristic . In this case, we can rely on the standard homomorphism from Q to F. Since this homomorphism extends to the respective polynomial rings, it follows that (9) holds over F[x , . . . , xm] when each polynomial on the right-hand side is replaced with its homomorphic image. (The images of the α i simply give the associated polynomials as we would consider them over F.) Hence, any assignment of a value in F to each of x , . . . , xm that causes every α i to vanish necessarily causes x x · · · xm to vanish as well, implying that at least one x i must have been assigned a value of . Since each realization A of the zero-nonzero pattern A corresponds to an assignment of a nonzero value to each of x , . . . , xm, it follows that no such realization can cause every α i to vanish, meaning that no such realization can have the characteristic polynomial x n . Now consider the case where F is a eld of characteristic q ≠ . Let M be the least common multiple of all the denominators of coe cients appearing in the polynomials p i . Then we have wherep , . . . ,pn ∈ Z[x , . . . , xm]. Let D be the nite set of prime divisors of M. When q ∉ D, we have that q is relatively prime to M. In this case, applying the standard homomorphism from Z to F to (10) shows that a nonzero multiple of x x · · · xm is contained in the ideal generated by α , . . . , αn in F[x , . . . , xm]. The remainder of the proof proceeds as in the characteristic case.
To say that a zero-nonzero pattern A is potentially nilpotent is to say that it allows all of the coe cients of the characteristic polynomial (other than the leading coe cient) to vanish simultaneously. It will be useful to consider a stronger condition that includes this as a special case.

De nition 4.3. Let
A be an n × n zero-nonzero pattern. We say that A is coe cient support arbitrary if for That is, to say that A is coe cient support arbitrary is to say that A allows the vector of coe cients (α , α , . . . , αn) ∈ F n to take on any support desired. In particular, then, its support can be empty, and so a zero-nonzero pattern that is coe cient support arbitrary is necessarily potentially nilpotent. But it is easy to see that the converse is not true; that is, a pattern can be potentially nilpotent without being coe cient support arbitrary. Of course, a pattern that fails to be coe cient support arbitrary cannot be spectrally arbitrary. Just as with the condition of being potentially nilpotent, there is an algebraic condition which, if satis ed, implies that the pattern is not coe cient support arbitrary. This condition is given by the next theorem, whose proof follows closely that of Theorem 4.2.
then A is not coe cient support arbitrary, except possibly over some elds of characteristic q ≠ , for nitely many values of q.
Proof. Suppose S ⊆ { , . . . , n} satis es condition (11). This means that there exists some k ∈ Z + such that for some polynomials p i ∈ Q[x , . . . , xm]. Let M be the least common multiple of the denominators of the coe cients appearing in the polynomials p i on the right-hand side of (12). Then we have for some polynomialsp i with coe cients in Z. Let D be the nite set of prime divisors of M. Let A be a realization of A over some eld F. Suppose for the sake of contradiction that when the variables x , . . . , xm are assigned values according to the corresponding nonzero entries of A, the support of the vector (α , . . . , αn) ∈ F n is equal to S. This means that α j is zero when j ∉ S and nonzero when j ∈ S.
When F has either characteristic , or has characteristic q ∉ D (so that q is relatively prime to M), the image of (13) under the standard homomorphism from Z to F gives an equation over F[x , . . . , xm] in which the left-hand side does not vanish, but the right-hand side does, when the variables take on those values as given by the entries of A. This contradiction shows that A cannot be coe cient support arbitrary over F.
A converse for Theorem 4.4 holds over C, due to this eld being algebraically closed.

Theorem 4.5. Let A be a zero-nonzero pattern. If
A is not coe cient support arbitrary over C, then A satis es (11) for some S ⊆ { , . . . , n}.
Proof. Let A be a pattern that is not coe cient support arbitrary over C; that is, suppose there is a set S ⊆ { , . . . , n} such that there is no realization of A for which the vector (α , . . . , αn) of associated polynomials has support S.
This means that there are no nonzero values for x , . . . , xm for which α i ≠ for i ∈ S while α i = for i ∉ S. In other words, any common zero of the polynomials in the set {α i : i ∉ S} must also be a zero of the polynomial x x · · · xm j∈S α j . Because C is algebraically closed, it follows from Hilbert's Nullstellensatz that This means that there exists some k ∈ Z + such that Note that the ideals in equations (14) and (15) Theorem 4.4 gives an algebraic condition, that of (11), whose utility seems somewhat one-sided. If the condition is satis ed for a particular pattern, then the pattern is not spectrally arbitrary. However, when the condition is not satis ed, this does not give any immediate conclusion. In contrast, it would be desirable to supply a purely algebraic condition that is equivalent to a pattern being spectrally arbitrary. (The analogous goal for sign patterns over R was stated as [14,Problem 14] in the unpublished notes of Shader.) In this section, we show that the question can be reframed in terms of the surjectivity of a related map that is simpler to study from an algebraic standpoint.

De nition 5.1. Let
A be an n×n zero-nonzero pattern over F, and let α , . . . , αn be the associated polynomials for A as de ned in (1). We de ne the polynomial map associated with A to be the map Fα : F m → F n de ned by Fα(x , . . . , xm) = (α (x , . . . , xm), . . . , αn(x , . . . , xm)). (16) In terms of De nition 5.1, the pattern A is spectrally arbitrary if and only if Fα maps (F \ { }) m surjectively onto F n . Considering (F \ { }) m as the domain of the function (as opposed to simply F m ) may complicate the application of algebraic (and, when F is taken to be R or C, analytic or topological) tools. However, the following lemma shows that it is possible to instead consider a related map that avoids this complication.

Lemma 5.2. Let
A be an n × n zero-nonzero pattern over F, and let Fα : F m → F n be the polynomial map associated with A. LetFα : F m+ → F n+ be de ned by x , . . . , xm). Then A is spectrally arbitrary over F if and only ifFα is surjective.
Proof. Suppose A is spectrally arbitrary. Then for each (a , a , . . . , an , a n+ ) ∈ F n+ there is a choice of values for x , . . . , xm such that f = a , f = a , . . . , and fn = an such that each x i is nonzero, allowing y to be chosen so that x · · · xm y = a n+ . Therefore,Fα is surjective. Now assumeFα is surjective. Then, for any a , a , . . . , an ∈ F, there exist x , . . . , xm , y ∈ F so that Fα(x , x , . . . , xm , y) = (a , a , a , . . . , an , ). In particular, x x · · · xm y = , implying that the x i are all nonzero. This shows that for each (a , a , . . . , an) ∈ F n there is a choice of nonzero x i at which the polynomial map Fα associated with A evaluates to (a , a , . . . , an). Therefore, A is spectrally arbitrary over F.
The question of whether a polynomial map such as theFα of Lemma 5.2 is surjective is one that can be studied from a variety of perspectives. For example, the tools of elimination theory from algebraic geometry allow one to gain information about this question by considering the polynomial ideal . . . , xm , y, c , . . . , cn , c n+ ].
One may ask whether the variety V(I) associated with this ideal, when projected onto its nal n + coordinates, gives all of F n+ . This question is equivalent to that of whetherFα is surjective, which by Lemma 5.2 is equivalent to the question of whether A is spectrally arbitrary. This reformulation of the question can be studied by considering the appropriate elimination ideal, many properties of which can be computed e ciently using Gröbner bases.

Veri cation of the n Conjecture for n ≤
Let A be an n × n zero-nonzero pattern, and recall that if A satis es condition (11) of Theorem 4.4 for some S ⊆ { , . . . , n}, then A cannot be coe cient support arbitrary, and hence cannot be spectrally arbitrary. For a xed such S, it is computationally feasible to use a computer algebra system to check this condition, at least when the number of nonzero entries of A is not large. Thus, in light of Conjecture 1.8, it is a natural idea to use such software to check, for a xed n and each possible S, whether this condition is satis ed for each pattern A with n − nonzero entries.
Of course, there is no reason to expect that every pattern that fails to be spectrally arbitrary in fact fails the condition given in (11). In particular, it might happen that a pattern is coe cient support arbitrary without being spectrally arbitrary. At the same time, it seems plausible that a pattern might have to satisfy (11) if it fails to be spectrally arbitrary for the reason that it is too sparse.
In any case, an exhaustive computation was performed using the SageMath software package [17]. To reduce the number of patterns to a feasible number, some avoidance of equivalent patterns was necessary. For zero-nonzero patterns, the appropriate notion of equivalence is provided by two observations. 1. Taking the transpose of a matrix does not a ect its characteristic polynomial. So two patterns may be considered equivalent when they are transposes of one another. 2. Applying a permutation similarity to a matrix does not a ect its characteristic polynomial, and two patterns are permutation similar if and only if they have the same digraph. So two patterns may be considered equivalent when their digraphs are isomorphic. The algorithm used in the computation was designed to exploit observation (2). Thus, an important feature provided by the SageMath software package was isomorph-free iteration over all digraphs of a xed order. In particular, the software includes an implementation of the "orderly generation" scheme of McKay [12] that allows the generation of every digraph on a xed number of vertices that has some property P, as long as property P is preserved by deleting edges. Of course, this does not hold for the property of having exactly n − edges. But the property of having at most n − edges is a suitable property with which to apply the generation scheme, and that is how the computer search proceeds.
The version of SageMath used did not allow the orderly generation of digraphs with loops. Therefore, the approach adopted was rst to generate every digraph on n vertices with m ≤ n − non-loop edges (since, for a pattern to be spectrally arbitrary, its digraph must have at least loops) and then, for each such digraph that was strongly connected (so as to give an irreducible pattern), to consider every way of assigning loops to some n − − m of the vertices.
For each digraph with loops so generated, a spanning tree was chosen (essentially at random) and the entries corresponding to the edges of this spanning tree were normalized to . Then the associated polynomials were computed and the condition given in (11) was checked. The complete search algorithm as implemented is outlined in Algorithm 1, while the actual source code used appears in the Appendix.  (11) is satis ed end if there is no S for which (11) is satis ed then record that A is 'exceptional' in that it may be spectrally arbitrary else record that A is not spectrally arbitrary (by Theorem 4.4) end end end end Algorithm 1: Procedure used for the exhaustive consideration of sparse n × n patterns using SageMath (see the Appendix for the source code) The results of the computer search are summarized in Table 1. Note that the number of patterns counted in the second column of the table is not, as would be ideal, equal to the number of inequivalent n × n patterns having n − nonzero entries and a digraph with at least two loops. In particular, although every such pattern is included in the search, some patterns are generated more than once. This is the case for two reasons. First, Table 1: Results of exhaustive search of irreducible patterns with n− nonzero entries using Algorithm 1. Note that the number of patterns generated is not equal to the number of inequivalent such patterns (see discussion).
n Number of patterns generated Number satisfying (11) for some S , , , , , , unknown the algorithm is not able to avoid considering a pattern and also its transpose, which corresponds to the same digraph but with every edge reversed. In addition, it may happen that di erent ways of adding loops to the "base" digraph D in Algorithm 1 result in isomorphic digraphs; in this case, each one will be considered separately and contribute separately to the count given in the table.
As the table shows, for n ≤ , every pattern generated was found to satisfy condition (11) for some set S. By Theorem 4.4, this implies that when n ≤ , every irreducible n × n zero-nonzero pattern with n − nonzero entries fails to be coe cient support arbitrary over every eld of characteristic . (The theorem also implies that no such pattern is coe cient support arbitrary over any eld of characteristic p ≠ , except possibly for nitely many values of p.) Combined with Theorem 1.10, this shows that the Generalized n Conjecture (i.e., Conjecture 1.8) holds for n ≤ , except possibly over some in nite elds of characteristic p > , for nitely many values of p.
The above observations would apply for n = as well, except that for exactly of the , patterns considered by the search algorithm, the condition given in (11) is not satis ed for any S ⊆ { , , . . . , }. Among these patterns we nd two distinct pairs of a pattern and its transpose. Hence, after taking into account equivalence, there are only di erent patterns to consider. Moreover, of these di er only in the placement of one of the loops in such a way that happens not to a ect the associated polynomials. Hence, there are in e ect only two di erent × patterns that could be counterexamples to Conjecture 1.8 over a eld of characteristic or q ≠ , except possibly for nitely many values of q. It follows that for n = it su ces to consider only signings of these as potential counterexamples to the original Conjecture 1.7. We present these patterns in detail in the next section, where we show that neither one is spectrally arbitrary over R. Hence, in fact no signing of either one can give a counterexample to Conjecture 1.7. Therefore, the original n Conjecture does in fact hold for n ≤ . Meanwhile, the above observations show that the stronger Generalized n Conjecture holds for n ≤ , except possibly over some in nite elds of characteristic p ≠ , for nitely many values of p. We may also consider the computational feasibility of performing the same veri cation for n = . For example, running the iteration without the expensive check of (11) reveals that the process would require checking , , digraphs. Extrapolation based on the length of time the process took to check the rst , of these suggests that this would require perhaps months of computing time. (The search for n = required a little over one day.) In any case, based on the outcome for n = , it seems reasonable to suspect that for n = a much larger number of n × n patterns exist for which the condition given in (11) is insu cient to resolve the question of whether or not they are spectrally arbitrary.
When considering a zero-nonzero pattern for which the condition given in (11) fails to give a conclusion (i.e., one for which the condition is not satis ed for any subset S), there seems to be an absence of e ective tools for determining whether or not the pattern is spectrally arbitrary. This situation is already in evidence for the very small number of × such patterns, as we will see in the next section. The exceptional patterns for n = Section 6 outlined an exhaustive search that found rst that for n ≤ no n × n zero-nonzero pattern is coecient support arbitrary over any eld of characteristic or of characteristic p ≠ , except possibly for nitely many values of p, and also that for n = this is true for all but a very small number of patterns. In fact, in order to understand every exceptional case found by the search, it su ces to consider only two patterns. We now analyze these and present what we know and do not know about their properties.
Example 7.1. The rst pattern found by Algorithm 1 not to satisfy (11) for any S ⊆ { , . . . , } is that with the digraph shown in Figure 2.
Following Theorem 2.6, the edges in Figure 2 are labeled such that those belonging to a particular spanning tree are normalized to have weight , while the weights of the remaining edges are a, b, . . . , g. This allows us to express the associated polynomials more succinctly; they are as follows.
We now consider the question of which values k ∈ R have the property that some matrix with this pattern has the characteristic polynomial x + kx. For such a matrix, we have α i = for all i ≠ . Then, to begin with, α = implies that d = −c. Substituting for d in the expression for α , we nd that α = implies = −c − a, so that then a = −c . Similarly, substituting d = −c in the expression for α shows that α = requires = −g − e, so that g = −e. We now consider α = and use the conditions derived above that d = −c, a = −c , and g = −e to obtain = α = −gbf − cda + ce + g(c + d) Hence, due to the condition that c ≠ , we have e = c . Finally, α = gives = dg(bf − c). Under the condition that dg ≠ , this gives c = bf . In fact, we can now express all variables in terms of b and f . In particular, we have: Substituting all of the above into α , and letting y = b f , we can write In particular, given that b fy = b f ≠ , the assumption that α = implies that y = − or y = − . Finally, we have not assumed any conditions on α , but we can still write it in terms of b, f , and y, as we did for α above. In particular, we have Since we have either y = − or y = − , the value of y(y + )( − y) is either or . Hence, by the above, we have either α = or α = b . In either case, we must have α ≥ .
Hence, the polynomial x + kx is not realized by any matrix over R with this pattern when k < . In particular, the pattern is not spectrally arbitrary over R.
Example 7.2. The other pattern found by Algorithm 1 that did not satisfy (11) for any S ⊆ { , . . . , } is that with the digraph shown in Figure 3. The edges in Figure 3 are labeled with those belonging to a particular spanning tree normalized to have weight , while the weights of the remaining edges are labeled, in rowmajor order, as a, b, . . . , g. This allows us to write the associated polynomials as follows.
To show that A is not spectrally arbitrary over R, we can apply Lemma 5.2 by showing that the mapFα given byF α (a, b, c, d, e, f , g, y) = (α , α , α , α , α , α , α , abcdefgy) (17) does not map R surjectively onto itself. By experimental computation, H. Tracy Hall [10] discovered a particular polynomial not realized by A, one with integer coe cients and terms. One of the referees was able to nd a simpler polynomial that A also fails to realize, namely x + x − .
In particular, using a Gröbner basis calculation to compute the intersection of this ideal with C[a] shows that the ideal given by this intersection (i.e., the corresponding elimination ideal) is generated by a + a + a + a + a + a + .
In any case, it is even more straightforward to use software to check that the above polynomial is contained in the ideal (18). This alone is enough to imply that if a through g and y are given values such that (α , . . . , α , abcdefgy) = ( , , , , , , − , ), then the polynomial (19) must vanish at these values. But this is not possible when a ∈ R, as can be veri ed by a simple computation via an application of Sturm's Theorem; see [1,Theorem 2.50]. Hence, the pattern corresponding to this digraph is not spectrally arbitrary over R.

Conclusion
Building on the results of [4] and [5], our work here shows that Conjecture 1.7, the original n Conjecture, holds for n ≤ . We also have established that Conjecture 1.8, the Generalized n Conjecture, holds for n ≤ over every eld, except possibly for in nite elds of nonzero characteristic p, for nitely many values of p.
Moreover, for such elds we know that there are only two patterns, given in Examples 7.1 and 7.2, that could be counterexamples for n = , and that these are not counterexamples over R.
For n = , our algorithm would need to check , , patterns, which is computationally not feasible at this time. It is possible that a partial run of our algorithm with n = will yield a pattern that is spectrally arbitrary, but further techniques would be needed to show that it is spectrally arbitrary.
Our algorithm relies on the notion of a coe cient support arbitrary pattern. The two patterns explored in Section 7 are coe cient support arbitrary over C, but we do not know if these are in fact spectrally arbitrary over C. Meanwhile, the authors were able to use Mathematica to check speci c polynomials to verify that each of the two patterns is coe cient support arbitrary over R as well, although in Section 7 we saw that these patterns are not spectrally arbitrary over that eld. Hence, being coe cient support arbitrary is in general not su cient to imply that a pattern is spectrally arbitrary. Over R, it is possible to de ne coe cient sign arbitrary in a natural way analogous to the notion of coefcient support arbitrary. While the conclusion of Example 7.1 shows that the pattern considered there is not coe cient sign arbitrary, computations similar to those mentioned above were able to show that the pattern of Example 7.2 in fact is. Hence, over R, even being coe cient sign arbitrary is not su cient to imply that a pattern is spectrally arbitrary.
Ultimately, new tools are needed in order to show that a pattern with n − nonzero entries is spectrally arbitrary. The Nilpotent Jacobian Method laid out in [14] and the Nilpotent Centralizer Method laid out in [9] both require the pattern to have at least n nonzero entries, as shown in [2]. Given appropriate new tools or methods, it may be possible to show that, as we suspect, the Generalized n conjecture is not true over C, and it even seems plausible that the two patterns presented in Section 7 are counterexamples.