Subalgebras in K[x] of small codimension

We introduce the concept of subalgebra spectrum, Sp(A), for a subalgebra A of finite codimension in K[x]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {K}[x]$$\end{document}. The spectrum is a finite subset of the underlying field. We also introduce a tool, the characteristic polynomial of A, which has the spectrum as its set of zeroes. The characteristic polynomial can be computed from the generators of A, thus allowing us to find the spectrum of an algebra given by generators. We proceed by using the spectrum to get descriptions of subalgebras of finite codimension. More precisely we show that A can be described by a set of conditions that each is either of the type f(α)=f(β)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(\alpha )=f(\beta )$$\end{document} for α,β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha ,\beta$$\end{document} in Sp(A) or of the type stating that some linear combination of derivatives of different orders evaluated in elements of Sp(A) equals zero. We use these types of conditions to, by an inductive process, find explicit descriptions of subalgebras of codimension up to three. These descriptions also include SAGBI bases for each family of subalgebras.


Example 2 Let be a primitive root of order 8.
Example 3 Let be a primitive root of order 12.
Example 4 Let be a primitive root of order 3.
It is not difficult to verify directly that we really get subalgebras. One can check that in fact, if given by generators, they are: We want to find general principles for how descriptions using conditions in our examples relate to descriptions in terms of generators and other characteristics of subalgebras.
We restrict ourselves to unital subalgebras of finite codimension n and give a classification for small n. Note that a unital subalgebra in [x] is commutative, associative and contains all constants.

SAGBI bases and type
One of our aims is to get a deeper understanding of the structure of SAGBI bases, for example to find ways to add an extra element to a SAGBI basis in ways that result in a new SAGBI basis. For this reason we remind the reader of some definitions, which we adapt to our univariate situation. More general definitions can be found for example in [1,2] or [3].
If A is a subalgebra in [x] the set S of all possible degrees of the non-constant polynomials in A form a numerical semigroup (that is an additive semigroup consisting of positive integers). It is well-known that such a semigroup is finitely generated. For any finite generating set we can find a finite set of polynomials G such that our set is exactly {deg g i |g i ∈ G}. We call G a SAGBI basis for A. A proper subset of G can be a SAGBI basis itself, but if there are no such subsets we say that G is minimal.
For any non-constant polynomial f of degree s ∈ S we can find a product g = ∏ g i ∈G g c i such that deg g = ∑ c i deg g i = s. Forming f − g with a suitable constant ∈ we can obtain a polynomial of smaller degree. We call this operation subduction. If the degree of the obtained polynomial still belongs to S, then we can perform another subduction. The importance of the SAGBI basis lies in the fact that f ∈ A if and only if there exists a sequence of subductions reducing f to a constant.
◻ Note that if d > 1 then the subalgebra A is contained in [x d ] and therefore it has infinite codimension. Such A are outside the scope of our work.

Subalgebras of codimension one
Next, let us look at subalgebras of codimension one (in [x]). Although relatively simple, these algebras give some insight. Obviously such subalgebras have type (2, 3) thus they contain polynomials of degree 2 and 3, which generate our subalgebra. Using variable substitution we can restrict ourselves to the case where the polynomial of degree two is x 2 . (Note that all constants are always in any subalgebra). Now the polynomial of degree three can be chosen as x 3 − ax . (Again, the constants are not essential and bx 2 can be subtracted). If a = 0 then we get a monomial case and know how to describe it from Theorem 1.
If a ≠ 0 then the replacement x → x with 2 = a reduces the situation to the case x 3 − x. So it is sufficient to study the subalgebra A = ⟨x 3 − x, x 2 ⟩. Note that for each odd k > 1 we have Proof We only need to recover the old variable. Then the monomial case corresponds to the first case and f (1) = f (−1) to the second. ◻ The above theorem already displays some ideas that we will try to generalise later on.

Derivations
Definition 1 Let ∈ . A linear map D ∶ A → is called an -derivation if it satisfies the condition for any f (x), g(x) ∈ A. We simply call it a derivation if it is an -derivation for some .

Subalgebras in K[x] of small codimension
A simple example is trivial derivation f (x) → cf � ( ). For A = [x] we have only trivial -derivations (with c = D(x) ), but as we will see later we can find other derivations in proper subalgebras. Note that the set of -derivations is a vector space over , but the set of all derivations on A is not. Nevertheless it is important for the future to note that a -derivation is also an -derivation if f ( ) = f ( ) for any f (x) ∈ A. Now we can formulate an important result obtained in [4], that will turn out to be pivotal for our continued exploration. Note that in [4] derivations are defined in a more general way, by the condition D(fg) = D(f ) (g) + (f )D(g), for some ring homomorphism ∶ B → . But in the same article it is shown that any homomorphism A → can be lifted to a homomorphism B → . Induction over codimension shows that in our situation such an algebra homomorphism is simply a homomorphism [x] → which is nothing else than a map f (x) → f ( ) for some ∈ . For that reason we can use -derivation in our reformulation.

Theorem 4 Any subalgebra A of codimension
We are thankful our referee for the following important remark. We can introduce more general ( , )-derivations as maps D ∶ B → such that and still have A = ker D as a subalgebra. Thus we can consider the map f (x) → f ( ) − f ( ) as ( , )-derivation and -derivations as ( , )-derivation which explains why many of our proofs below are quite similar for both alternatives. In fact his comments give much deeper generalisation, but we restrict ourselves by this remark only.

Subalgebra conditions
A straightforward induction argument using Theorem 4 shows that any subalgebra A of codimension n can be described by n linear conditions L i (f ) = 0 where L i is either a derivation of some subalgebra containing A or has the form Our main hypothesis when initiating this work (which will be proved later) was that linear conditions defining subalgebras can be stated in a neater way. Namely, we hoped that for any subalgebra of finite codimension m there would exist a finite set, which we will call the spectrum of the algebra, and m linear conditions expressed in terms of f(x) and finitely many derivatives f (k) evaluated in the elements of the spectrum which determine if f (x) ∈ A. We have seen such conditions in Theorems 1 and 3 and in Examples 1-4 and want to understand their nature.
We want them to be subalgebra conditions. By this we mean that the set of all polynomials satisfying the conditions form a subalgebra. Since our conditions are linear we only need to demand two things for them to be subalgebra conditions. Firstly, a trivial one: that constants should satisfy the conditions. Secondly, a nontrivial one: that whenever f(x) and g(x) satisfy the conditions, so does the product f(x)g(x).
For example the condition f ( ) = 0 is not a subalgebra condition, because the non-zero constants do not satisfy it. But the condition f ( ) = f ( ) is a subalgebra condition. The same is true for the condition f � ( ) = 0.
The single condition f � ( ) + f � ( ) = 0 is not a subalgebra condition, but together the conditions f ( ) = f ( ), f � ( ) + f � ( ) = 0 are subalgebra conditions. As this example shows being subalgebra conditions is a property of a set of conditions. (The set may, however, as in the first two examples, consist of just one element.) In general, any condition gives a set of subalgebra conditions. Indeed since the conditions are linear we only need to check that if f(x) and g(x) satisfy the conditions then the same is true for f(x)g(x). We have One can find generalisations including derivatives of higher order, but we skip this for now and show only one spectacular example of subalgebra conditions:

Spectrum
Now we want to introduce the main definition of this article.

Definition 2 Let
A be a subalgebra of finite codimension. Its spectrum consists of ∈ such that either f � ( In the second case obviously belongs to the spectrum as well. We write Sp(A) to denote the spectrum of the algebra A.
Unfortunately the word spectrum already has a specific meaning, so it would be more correct to use something like "subalgebra spectrum", but because we believe that this notion is very important and that the word spectrum reflects this concept very well we use the word "spectrum". This makes our article more readable and in our context the interpretation should be unambiguous.
We have already seen in Theorem 3 how the spectrum naturally arises in the description of subalgebras of codimension one.
One trivial but useful remark is the following. Proof Induction and Theorem 4 shows that A is a subalgebra of a subalgebra of codimension 1. Then Theorems 5 and 3 finish the proof. ◻ While this theorem is valid for any proper subalgebra in [x], it is shown in Sect. 27 that for polynomials in two variables one can find a proper subalgebra with empty spectrum.
One of our main results can be formulated as follows.

Theorem 7 If A is a proper subalgebra of finite codimension then only the values of f(x) and finitely many of its derivatives
We will prove this later. We already have done so for monomial subalgebras and for subalgebras of codimension one.
Before moving on we give some equivalent definitions of the spectrum.
Theorem 8 Let A be a subalgebra of finite codimension and ∈ . The following is equivalent.
(i) belongs to the spectrum of A.
belongs to the spectrum of the subalgebra ⟨p(x), q(x)⟩ for each pair of monic p(x), q(x) ∈ A with relatively prime degrees.
Note that the condition of relatively prime degrees in (iv) is necessary since it guarantees that ⟨p(x), q(x)⟩ is of finite codimension.
Proof (ii) is a simple reformulation of (i). (Note that we can take = when the condition is f � ( ) = 0).
(ii) implies (iii) almost directly. We choose any SAGBI basis and replace each element g by g − g( ) obtaining a new SAGBI basis.
(iii) implies (ii) because any f (x) ∈ A can be subduced to a constant c. In each subduction step a polynomial divisible by It is easy to see that we must have c = f ( ).
By Theorem 5 (i) implies (iv). The opposite, that (iv) implies (i) is more difficult.
If there exists f (x) ∈ A such that f � ( ) ≠ 0 we need to find . Subtracting a constant we can suppose that f ( ) = 0 and let 1 , … , k be the other roots of f(x), which exist because A is a proper subalgebra and is algebraically closed. Then should equal some i . If the implication does not hold then for each i there exists g i (x) ∈ A such that g i ( i ) ≠ g i ( ). Subtracting a constant we can suppose that g i ( ) = 0, but g i ( i ) ≠ 0. Now, using that our field is infinite, we can easily construct a linear combination g(x) of the g i , such that g( ) = 0 but g( i ) ≠ 0 for each i. Since A has a finite codimension we can for each large degree find a polynomial that belongs to A. We choose such a monic polynomial h(x) that has degree larger than deg g(x) and relatively prime to deg f (x). We can also suppose that h( ) = 0.
The next step is to construct a polynomial p(x) = h(x) + cg(x) that has the same property as g(x), namely p( ) = 0 but p( i ) ≠ 0 for each i. Again, this is possible because our field is infinite. Let q(x) be f(x) divided by its leading coefficient. Consider the subalgebra ⟨p(x), q(x)⟩. Because belongs to its spectrum and q � ( ) ≠ 0 there exists such that p( ) = p( ) and q( ) = q( ). But q( ) = 0 so = i for some i. On the other hand 0 = p( ) ≠ p( i ) and we get a contradiction. This proves that our assumption that (iv) does not imply (i) must have been wrong. ◻

Linear independence
To be sure that the maps we use later are linearly independent we need to prove some auxiliary statements, even though they may seem quite obvious or well-known. Then L ij (f ) = 0 for j > 1 and for j = 1, i < k. On the other hand and we get a contradiction. ◻ .

Subalgebras in K[x] of small codimension
Naturally we can have some dependencies in a proper subalgebra but we want to show that all of them are linear combinations of defining subalgebra conditions. Theorem 10 Let V be a vector space over and L 1 , … L n be linear independent linear maps Proof We use induction on n. If n = 1 then L 1 ≠ 0 and we can choose a vector v ∈ V such that L 1 v = 1. Then V = v + ker L 1 and because ker L 1 ⊆ ker l we get that ker L 1 and the arguments above shows that l| U = c 1 L 1 | U . Applying the induction to l − c 1 L 1 considered on U we get that l − c 1 L 1 = ∑ n i=2 c i L i and we are done. ◻

The size of the spectrum
How large can the spectrum of a subalgebra of finite codimension n be? To answer this question we first prove an important statement, which essentially says that elements in the spectrum appear in a natural way and there are no "ghost" elements in the spectrum. Otherwise the extra condition is an -derivation for some that does not belong to Sp(B). By Theorem 18 below we can replace it by f (x) → f � ( ) and it remains to rename to 0 . ◻

The characteristic polynomial for subalgebras on two generators
Now we want to understand how to find the spectrum. We start from a special case where we can explicitly construct a polynomial which roots are exactly the elements of the spectrum. Let p(x), q(x) be two monic polynomials. Consider the following polynomials in two variables: We now introduce a definition that will be helpful when determining the spectrum of the subalgebra generated by p and q.

Definition 3
The characteristic polynomial p,q is the resultant of polynomials P and Q considered as polynomials in y.
Its roots are 1 and −1 and this gives some insight into why f (1) = f (−1) was the subalgebra condition for A = ⟨x 3 − x, x 2 ⟩.
It is easy to check that get x 3 ,x 2 (x) = x 2 and this can be easily generalised, as shown below.
Proof Assume without loss of generality that n > m . First note that the polynomials

This means that
If m = 1 , this determinant is upper triangular and equal to 1 = x (m−1)(n−1) . This will be the base case for a proof by induction.
will have x m as first nonzero element, in column m + i . One can break out a factor x m from each of these rows. Now where A is an upper triangular (m − 1)-matrix with ones on the main diagonal. Expanding the determinant along the first column m − 1 times and rearranging the rows gives which is of size (n − 2) . Note that this is exactly the characteristic polynomial of

Computing p,q
An alternative proof of Theorem 13 stems from computing the resultant Res y (P, Q) using reductions of P by Q and vice versa. A more exact statement is given in the below proposition. Here lc(Q) denotes the leading coefficient of Q(x, y) when regarded as a polynomial in y with coefficients in [x] , and degrees are also taken with respect to y.

Similarly we have that
Proof This follows from the fact that the resultant where i are the roots of Q when regarding Q as a polynomial in y. (See [5].) The roots are counted with multiplicity and they may lie in some extension of the field [x] of coefficients. Now, in the same way, we get an expression and by comparing the two expressions we get the first statement of the proposition. The second statement is proven in the same way, but is slightly easier to handle since Res y (Q, P) = lc(Q) deg(P) ∏ P(x, i ) does not contain powers of −1 . ◻ The above proposition, which also can be found in [6], can be a useful tool for computing the characteristic polynomial when the generating polynomials are sparse. Let us first look at the two easy examples. We have and The second computation can be generalised to any case of two monomial generators, as follows. If p = x n and q = x m with n = qm + r we can subtract powers of y

Note that the latter sum equals H(x, y) = r(x)−r(y)
x−y for h = x r . This can be used to obtain a more elegant inductive proof of Theorem 13 as By the induction hypothesis q,h = x (m−1)(r−1) and this results in as we already knew from Theorem 13.
Finally, let us consider a more complicated example where we compute the char- is formed by subtracting multiples of Q from P in such a way that the y-degree decreases. Now deg y (P 1 ) = 3 and deg y (Q) = 4 so the next step is to form In the next step we want to reduce P 1 using Q 1 , but Q 1 has a non-constant leading coefficient a(x) = x 5 + x 2 + 4 . We can get around this problem as follows: is our desired resultant. This shows that Sp(A) has 16 elements. (One can verify that h(x) has no multiple roots.) Res y P 1 , Q = lc(P 1 ) 2 Res y P 1 , Q 1 = Res y P 1 , Q 1 .
As the above example shows, even if the process of computation is fairly simple, it is not easy to track how the resulting polynomial relates to the initial polynomials.
There are more efficient methods for computing resultants. For computing a resultant of two bivariate polynomials p, q of degree at most k there are well-know algorithms with time complexity k 3+o (1) . There are also more efficient algorithms known, see [7], but for our purposes standard implementations have been efficient enough.

Properties of p,q
Let us next turn to an interesting property of p,q . The below theorem relates p,q to partial derivatives of a polynomial F that turns up when applying the standard algorithm for computing a SAGBI basis for ⟨p, q⟩ . (See [1].) As we will see this theorem turns out to be an important building block for showing that derivations of non-spectral elements are trivial.
Proof Let us look at what we did in the proof of Theorem 13 again. In a complete expansion of the determinant we choose in each column j either x j−i (if we choose row i from the first m − 1 rows ) or we choose x j−i+(m−1) (if we choose a row i between the last n − 1 rows). Because ∑ j = ∑ i we get a total degree in the product equal to (n − 1)(m − 1). We can never get larger degree. The difference when we use p(x) and q(x) instead is that we add some terms of smaller degree in each element of the matrix. But they cannot affect our maximum total degree term x (n−1)(m−1) so the highest coefficient in p,q (x) at x (n−1)(m−1) is the same as for the monomial case.
To prove the second statement we use a well-known fact (see [8]) that where the product is taken over all roots of q(y) − q in some field extension with multiplicity. When we evaluate this for p = p(x) and q = q(x) we get zero because y = x is one of the roots. If we take a partial derivative with respect to p first and evaluate in p = p(x) and q = q(x) after that we get a sum over roots where all terms except one (corresponding the root y = x ) are zero. But we can get this remaining Now, using another property of the resultant we get where all resultants above are evaluated in y. Here we have also used that for any We obtain the second formula in a similar way and the signs should be As expected we get: Are there instances when f ,g has an infinite number of roots? The answer is yes, as we have already seen for certain monomial subalgebras. More precisely we have seen that x m ,x n = 0 if and only if (m, n) > 1 . We will now generalise this result.

This means that P(x, y) = p(x)−p(y)
x−y has a factor h(x)−h(y) x−y which is a polynomial in y of degree at least one. Similarly if q(x) ∈ [h] then Q(x, y) also has this factor so they have a common factor as polynomials in y over (x) and as a consequence their resultant p,q (x) is equal to zero.
To prove the opposite assume now that deg p(x) = n and deg q(x) = m . Let F(p, q) be the resultant of p(x) − p, q(x) − q , as before. We know from lemma 19 in [1] that F(p, q) = ∑ in+jm≤nm c ij p i q j where c ij are constants in . Moreover, it follows from that lemma that p m has non-zero coefficient and all other terms contain p to a power strictly lower than m. Assume now that p,q (x) = 0 . Then it follows from Theorem 14 that we can differentiate F with respect to p and get another identity involving p and q. Regarding p as variable this identity is a polynomial of degree m − 1 with coefficients in (q) , showing that adjoining p to the field (q) is an extension of degree at most m − 1 . From Lemma 13 in [1] we get the first On the other hand we know by Theorem 14 in [1] that (p, q) = (h) for some polynomial h and this means that we have a polynomial h of degree

How the spectrum relates to p,q (x)
Now we want to compare the roots of the characteristic polynomial with the spectrum.
To start with we will focus our attention on a special case -an algebra A generated by two monic polynomials p(x), q(x) of degrees m > n with (m, n) = 1. It is known that they form SAGBI basis for A (see [1]) and therefore A has codimension g(m, n) = (m − 1)(n − 1)∕2 . (Here g(m, n) is the genus of the corresponding semigroup of degrees.) So if we want to describe this algebra we need to find g(m, n) subalgebra conditions. For m = 3, n = 2 we have done that in Theorem 3.

Theorem 16 Let
The following is equivalent.
is a root of the characteristic polynomial of A.
Proof The alternatives (i) and (ii) are equivalent since each of the two conditions stated in (ii) are closed under sums and products, so we need only to prove that (ii) and (iii) are equivalent. By the fundamental property of the resultant (see e.g. [8]) we know that is a root of the characteristic polynomial if and only if there is some ∈ such that P( , ) = Q( , ) = 0.

Subalgebras in K[x] of small codimension
We now regard two different cases. The first case is when ≠ . In this case we have that p( ) − p( ) = ( − )P( , ) = 0 and similarly q( ) = q( ) . Thus the second statement of (ii) holds.
The other case is that = which means that 0 = P( , ) = p � ( ) . (The second equality can easily be derived from the definition of P as P(x, y) = (p(x) − p(y))∕(x − y) .) In the same manner we find that q � ( ) = 0 so in this case the first statement of (ii) holds. ◻ This shows that the characteristic polynomial allows us to find the spectrum explicitly, for the subalgebras we currently study. Note that the theorem also shows that the characteristic polynomial is never a constant, because the spectrum is always non-empty.
Also note that Theorem 8 gives us a theoretical way to find the spectrum for any subalgebra. In most practical cases it is sufficient to consider only p,q for each pair {p, q} of generators, but the problem is that their degrees are not always relatively prime.
Here is another application of the theorem.

Proposition 2 If a(x) is a polynomial of degree at least two that divides both p(x) and q(x) then all the roots of a(x) are roots of p,q (x).
Proof This shows that common factors of p and q with no multiple roots are also factors of p,q . By modifying the proof a little we can get the same result also for factors with multiple roots.

Theorem 17 If a(x) is a polynomial of degree at least two that divides both p(x) and q(x) then a(x)| p,q (x).
Proof Assume that p(x) = a(x)p 1 (x) and q(x) = a(x)q 1 (x) . Now we modify a(x) to separate its roots by introducing an additional variable s.
Let p =ãp 1 and q =ãq 1 . It follows that P (x, y) = P(x, y) + sR(x, y) and Q (x, y) = Q(x, y) + sT(x, y) for some polynomials R, T with coefficients in k[s]. Thus for some polynomial h. The first equality in the last row comes from the fact that when computing the resultant from its definition as a determinant, every term that has been added contains at least one factor s. Now P ,Q (x) has a factor ã(x) by the previous theorem. That is P,Q + sh(x, s) =ã(x)u(x, s) . Now let s = 0 to conclude the proof. ◻ P,Q (x) = Res y (P(x, y) + sR(x, y), Q(x, y) + sT(x, y)) = Res y (P(x, y), Q(x, y)) + sh(x, s) = P,Q + sh(x, s) As an example, let us apply the above theorem as a tool for finding conditions for the subalgebra A = ⟨x 4 − x 2 , x 3 ⟩.
We see that x 2 divides both generators so it should divide the characteristic polynomial as well. Thus zero is in the spectrum. Moreover f � (0) = 0 is valid for both generators and therefore is one of the conditions. Because g(4, 3) = 3 we should find two additional subalgebra conditions. The characteristic polynomial can be found using Maple and it is equal to x 2 (x 4 − x 2 + 1).
Thus, besides zero we have four other elements in the spectrum, which are in fact primitive roots of degree 12. If we name one of them , the remaining ones will be 5 , 7 , 11 . From Theorem 12 we find that the remaining conditions are of equality type. Thus we need to arrange those primitive roots in pairs to get conditions of the form f ( ) = f ( ). It is not hard to check that In fact, experiments suggests that when the degree of the factor a(x) is higher than two its multiplicity as a factor of the resultant is higher.

Conjecture 1 If a(x)
is a polynomial of degree at least two that divides both p(x) and q(x) then a(x) deg(a)−1 | p,q (x).

Derivations in A
Now we want to formulate some general statements about derivations. We begin our study with a subalgebra A, generated by two polynomials p(x) and q(x) of relatively prime degrees. As we know (see [1]) p(x), q(x) form a SAGBI basis and has one relation F(p, q) = 0 arising from the corresponding resultant. (Thus this F is the same as in theorem 14.) Denote D(p(x)) = Dp and D(p(x)) = Dq . First note that for any polynomial G(p, q) we have If we denote F p (p( ), q( )) by F � p ( ) and F q (p( ), q( )) by F � q ( ) then we get that is a necessary and sufficient condition for a linear map D to be a derivation of A. Note also that taking the ordinary derivative in we get Suppose now that does not belong to the spectrum Sp(A). Then we know that the vector v = p � ( ), q � ( ) is non-zero. Also, according to Theorem 16, we have p,q ( ) ≠ 0. Now it follows from Theorem 14 that the vector w = F � p ( ), F � q ( ) ≠ (0, 0).
As the above equalities show that both the non-zero vector v = p � ( ), q � ( ) and the vector (Dp, Dq) are orthogonal to w, they must be parallel. This means that (Dp, Dq) = c p � ( ), q � ( ) and thus we simply have D(f (x)) = cf � ( ) . In other words, D is a trivial derivation. Now we generalise this to an arbitrary subalgebra A.

Theorem 18 Let A be an arbitrary subalgebra of finite codimension and D be an
. Subtracting a constant we can suppose that g i ( ) = 0, but g i ( i ) ≠ 0. Beside that there exists g 0 (x) ∈ A such that g 0 ( ) = 0, but g � 0 ( ) ≠ 0 (all this because does not belong to the spectrum). Now, using the fact that an algebraically closed field is infinite, we can easily construct a linear combination g(x) of the g i such that g( ) = 0 but g( i ) ≠ 0 for each i > 0 and g � ( ) ≠ 0. Since A has a finite codimension we can for each large degree find a polynomial of that degree that belongs to A. We choose such a monic polynomial h(x) that has degree larger than deg g(x) and relatively prime to deg f (x). We can also suppose that h( ) = 0.
Our next step is to a construct polynomial p(x) = h(x) + cg(x) that has the same property as g(x), namely p( ) = 0, p � ( ) ≠ 0 and p( i ) ≠ 0 for each i > 0. Again, this is possible because our field is infinite. Let q(x) be f(x) divided by its leading coefficient. Consider subalgebra B = ⟨p(x), q(x)⟩. By construction does not belong to its spectrum, so according to our arguments before the theorem the restriction of D to B should be a trivial derivation therefore D(f (x)) = cf � ( ) = 0 which is a contradiction.
The rest is easy. Any polynomial in A can be written as a linear combination of g 0 (x) , some constant and some polynomial f(x) having as double root. Therefore only the value of g 0 (x) determine the value of D, so it is sufficient to find c such that D(g 0 (x)) = cg � 0 ( ). ◻

Clusters
Let us now introduce a natural equivalence. For a given algebra A we define ∼ if and only if f ( ) = f ( ) is valid for all f ∈ A. Then the spectrum of the subalgebra A is a disjoint union of equivalency classes that we call clusters. If A is obtained from B by a linear condition L(f ) = 0 then Theorem 11 gives us a simple connection between clusters in B and A.
If L is an -derivation then the clusters are the same if ∈ Sp(B) and { } constitutes an additional cluster in A if ∉ Sp(B).
If L(f ) = f ( ) − f ( ) there are several possibilities. If neither nor belongs to the spectrum of B then they together form a new cluster.
If exactly one of them (say ) belongs to the spectrum of B then we simply add to the cluster containing . At last if both and belong to the spectrum of B then they should lie in different clusters and as a result those two clusters will be joined in A.
From now on we will use the notion A(C) = {f (x)|f ( ) = f ( ) for all , ∈ C} for the subalgebra defined by the fact that all its elements have the same value on the cluster C.

The main theorem
Now we want to prove Theorem 7. We begin with the following statement. Proof We use the same notation as in Theorem 19. According to that theorem we have polynomials in A of each degree greater than Ns − 1. If we complete them to a

Theorem 19 Let A be a proper subalgebra of [x] with Sp(A) = { 1 , … , s } and let
linear basis in A we get a set Q, consisting of exactly Ns − n new polynomials q and we can suppose that 1 ∈ Q. Consider the vector space V consisting of linear maps We have that dim V = Ns . Consider its subspace W of those maps that annihilate all q ∈ Q. The subspace W has dimension n (because the condition D(q) = 0 is a homogeneous linear equation on the set of the coefficients c ij ). We choose a basis in W consisting of n maps D and claim that the conditions D(p) = 0 for each D from this basis describes A. Indeed those conditions by construction describe exactly the subspace generated by q ∈ Q in the subspace of all the polynomials of degree less than Ns. It remains to show that each x i N A is annihilated by D. Let D 0 be the map Because A ( j ) = 0 for each j we have that D 0 (x i N A ) = 0 and it is sufficient to consider D 1 = D − D 0 consisting of only the derivatives. D 1 annihilates all the elements of the form x i N A because it has derivatives of degree at most N − 1 and the same is true for D.
Thus our conditions are valid on all basis elements in A and describe the vector space they generate, which is A. In other words the conditions that E i (p(x)) = 0 for our basis elements E i ∈ W determine the subalgebra A. Note that this automatically implies that we get subalgebra conditions. ◻ In fact we can prove a stronger result. The subalgebra conditions are either of form f ( ) = f ( ) or derivations. For non-trivial derivations we can prove (see [9]): Theorem 21 If belongs to the spectrum then each -derivation D can be written as thus using pure derivatives (of some order) in the elements of the cluster containing .

One element in the spectrum
Now we want to show some applications of the spectrum. We start from the subalgebras which have only one element in the spectrum. .

Theorem 22
Let A be a subalgebra of codimension k ≥ 1. The following statements are equivalent.

A is defined by k linearly independent conditions of the form
Proof We can use induction on k. The base for the induction is guaranteed by Theorem 3. Let k ≥ 2. Using the change of variable x = x − we can restrict ourself to the case = 0.
(1) ⇒ (2). According to Theorem 4 the algebra A is obtained from B as a kernel of some linear map. This map should be 0-derivation D, otherwise we have more than one element in the spectrum. By Theorem 5, B should have zero spectrum and according to the induction hypothesis B contains some monomials x m , x n with (m, n) = 1. Note that m, n > 1 because B is a proper subalgebra. Using that (2) ⇒ (1). Because the subalgebra generated by x m and x n has spectrum zero, by Theorem 5 the spectrum of A cannot have any other elements than zero.
(3) ⇒ (2) All the monomials x m with m > N satisfy the conditions. ◻

Subalgebras containing a polynomial of degree 2
Suppose that the subalgebra A contains a polynomial q(x) of degree two. Two trivial cases are A = ⟨q(x)⟩ and A = [x]. In non-trivial cases we should have a polynomial p(x) of odd degree 2l + 1 ≥ 3. If we suppose that l is as small as possible then it is easy to see that A = ⟨p(x), q(x)⟩. Using variable substitution we can suppose that q(x) = x 2 . Subtracting even terms we can WLOG suppose that p(x) is an odd polynomial, thus for some monic polynomial a(x) of degree l. We want to show that the spectrum of A consists of the roots of a(x 2 ).

Theorem 23 Any proper subalgebra A of finite codimension in [x] containing a polynomial q(x) of degree two has a spectrum consisting of g elements for some
g > 0 . The spectrum has k = g 2 pairs { i , i }, i = 1, … , k such that for each i the sum i + i has a constant value 2 0 and (for odd g) one extra element, namely for odd g only).

Vice versa, if an algebra satisfies such conditions, then it is generated by
Proof Since the codimension is finite and the subalgebra is proper we can after substitution suppose that A is generated by As we already discussed above the spectrum has exactly g elements. To check the conditions note that they are trivial for x 2 and that if j ≤ m i for i > 0. If m 0 > 0 then all the derivatives until 2m 0 + 1 are zero as well. Therefore p(x) and q(x) satisfy the conditions and it is sufficient to check that if f(x) and g(x) satisfy the conditions the same is true for f(x)g(x). We have and get the desired property both for i > 0 and i = 0 (because if j is odd one of j 1 , j 2 is odd as well). So A satisfies the conditions. Let us now turn to the opposite direction. Our proof shows that the conditions determine some subalgebra that contains A and we need to prove that it equals A. If not there should be some polynomial f(x) which does not belong to A. Using subduction by p(x) and q(x) we can suppose that it has an odd degree less than the degree of p(x) and has only odd powers, and thus

Note that for an odd function f(x) we have
We get the opposite sign compared to our conditions so all terms must be zero. Thus i and i have multiplicity at least m i + 1 as zeroes of f(x). Similarly the second condition gives us that the multiplicity of zero as a zero is at least 2m 0 + 1. But then f(x) cannot have degree less than the degree of p(x).
It remains to understand how we get back to the general case by using variable substitution back. Obviously 0 is the only root of the derivative in q(x) and the spectrum is simply shifted by 0 . ◻

˛-Derivations
Now we want to collect some general properties of the set D A of -derivations. Let A be a subalgebra of finite codimension and M = {f (x) ∈ A|f ( ) = 0} be the corresponding maximal ideal in A.
Theorem 24 Let D ∶ A → be a linear map.
1. The following statements are equivalent: Proof (1) Obviously any -derivation has these properties so we need only to work in the opposite direction. For any f ∈ A we have that Now for any f , g ∈ A we have: (2) If we pick any SAGBI basis in M and choose those g i from it that form a basis modulo M 2 then D will be uniquely determined by the values of D(g i ).
On the other hand we get an -derivation for any choice of such values if we extend to a linear mapping that vanishes on constants and M 2 , by part (1). The values of D on the remaining elements in the SAGBI basis will be uniquely determined. This also proves (3) as we can start from a SAGBI basis with s elements and then possibly remove some of them to obtain our basis for the space of derivations. ◻ The type T(A) = (d 1 , … , d s ) gives us an upper bound d ≤ s and we have equality in the monomial case. Indeed it is easy to check that the maps D i ∶ f → f (d i ) (0) form a linear basis for the set D A 0 of zero-derivations. Now we consider the following chain of subalgebras that differ by one in codimension: We see that d can both decrease by one and increase by two.

Some applications
Now we want to show some applications of the spectrum. First of all the spectrum gives us a much clearer picture of the inclusion of one subalgebra in another.
For example, which subalgebras B of codimension 2 can contain the subalgebra A = ⟨x 4 , x 3 ⟩ of codimension 3? The subalgebra A ⊆ B has an element of degree 3, which does not belong to the semigroup generated by 2 and 5 so the type of B cannot be (2,5), thus T(B) = (3,4,5).
Also Sp(A) = {0} implies that Sp(B) = {0} and our only candidate for B in the classification obtained in Theorem 27 below is s = 1 with = 0. Using that x 3 ∈ B we can specify the conditions from the Theorem further to f � (0) = f �� (0) = 0 and hence B must be the monomial algebra ⟨x 3 , x 4 , x 5 ⟩.
Another obvious application is finding the intersection of two subalgebras: we take the union of their spectra and the union of their conditions and we only need to check if there are any linear dependencies between the conditions. For example we can easily spot the situations when the intersection of two subalgebras is a monomial subalgebra. Both should have zero spectrum and the conditions of the subalgebras should complete each other so that we obtain conditions of the form f (j) (0) = 0.
We can go in the opposite direction as well: if we have two subalgebras A 1 , A 2 we can easily construct the subalgebra they generate together. We take the intersection of the spectra and try to see which conditions remain. Let us take an example from [2]. Is ⟨x 3 − x, x 4 , The subalgebra ⟨x 4 , x 5 ⟩ is monomial, so its spectrum is zero. But zero is not in the spectrum of the subalgebra ⟨x 4 , x 3 − x⟩, so the intersection of their spectra is empty and we get [x]. The most important application is the possibility to construct SAGBI bases without having to invoke the standard algorithm based on subduction. We will expand on this aspect in the next section. ⟨x 4 , x 6 , x 9 , x 11 ⟩ ⊃ ⟨x 4 , x 6 , x 9 ⟩ ⊃ ⟨x 6 , x 8 , x 9 , x 10 , x 13 ⟩.

3 2Constructing SAGBI bases
One useful thing we want to mention is that the inductive approach which we have used throughout the article also allows us to create SAGBI bases in A relatively easily. Namely, when we have a SAGBI basis G for B and get A by adding the condition L(f ) = 0 we do the following to obtain a SAGBI basis of A. All elements of G that satisfy the extra condition L(f ) = 0 will remain in the SAGBI basis. There must, however, be at least one element that does not satisfy the condition. Let us choose such a g ∈ G of minimal degree d, thus L(g) ≠ 0. Note that exactly this degree d should disappear from the numerical semigroup S of degrees. Thus we know the new semigroup S A = S ⧵ {d} and can easily find the type (s 1 , … , s m ) of the subalgebra A. For each degree s i we find a polynomial h i ∈ B and our new SAGBI basis consists of f i = L(g)h i − L(h i )g . If we wish to make them monic we can just divide each f i by its highest coefficient. In order to further simplify calculations we want the basis elements to be inside M , and there are several ways to do this. The simplest one is to replace f i (x) by f i (x) − f i ( ), but a more efficient way is to choose h i and g in M from the start. Sometimes it may be clever to choose a linear combination with the previous f j to get as high a degree of the factor x − as possible.
We summarize this as follows.
Theorem 25 Let G be a SAGBI basis for B chosen inside M B . Let g = g i be an element of minimal degree in this basis that does not belong to A. Suppose WLOG that L(g) = 1 and L(g j ) = 0 for j ≠ i (which we can obtain replacing g j by g j − L(g j )g).
• The set consisting of polynomials g j , h j = gg j − L(gg j )g with g j ∈ G , j ≠ i and two polynomials f k = g k − L(g k )g for k = 2, 3 forms a SAGBI basis for A inside M A . (Not necessary a minimal one.) • If A has type (s 1 , … , s m ) then to construct a minimal SAGBI basis one should for each s = s j find a polynomial p s ∈ B of degree s and take p s − L(p s )g. If all p s are chosen inside M B then the obtained SAGBI basis will be inside M A .
This immediately proves the second statement because we get elements of degree s i in A. To prove the first statement we need to find polynomials built up from our basis elements of each degree d ≠ deg g occurring in B. We can express d as the degree of some g l u where u is a product of g j , where j ≠ i, but repetitions are allowed. Because each such g j belongs to M A the same is true for u, so suppose that l > 0. If l ≥ 2 we can use f a 2 f b 3 u where l = 2a + 3b to get the degree d. At last if l = 1 and u = g j v for some g j we can use h j v . ◻ Now we want to prove another result, where we want show how to use SAGBI bases.

3
We know that g k ∈ M B 2 for k ≠ i . As a result Using the facts that g 4 = (g 2 ) 2 , (g 2 )g j ∈ M A 2 and gM A ⊆ M A we find that Applying this for k = j in (1) we can improve this to From this it is clear that Thus the two -derivations of A we have found earlier form a basis for D A . ◻

SAGBI bases in codimension one
Let us see how to find SAGBI bases in subalgebras of codimension one. We start from [x] (which has x as SAGBI basis and from which we can get A either by the condition f � ( ) = 0 or by the condition f ( ) = f ( )). We now get Theorem 3 without any effort thanks to Theorem 4. Now we want to prepare for the next codimension and for this we need to find SAGBI bases and derivations for the different subalgebras of codimension one. We obviously have that D contains f �� ( ), f ��� ( ) in the first case and f � ( ), f � ( ) in the second case.
Because d and d are not greater than the number of generators, which equals two, we have found all nontrivial derivations.
Type (2,3) is the only possible semigroup of degrees, so an easy way to construct a SAGBI basis is to use the second part of Theorem 25. We get as the SAGBI basis for the first alternative, that is when L is a derivation. For the second alternative we can choose

Subalgebras of codimension two
We now turn to subalgebras of codimension two. By Theorem 4 they can be obtained by applying one extra condition to a subalgebra B of codimension one. This means we need to study how those conditions look. In the case when the extra condition is f ( ) = f ( ) we simply add one or two elements to the spectrum and obtain the algebra A. This is an easy case. A more difficult case is when we need to describe a kernel of some derivation. But we already know the derivations in each of the two cases considered above. Thus we are prepared to make a classification of all codimension two subalgebras: Theorem 27 Let A be a subalgebra of codimension two. Then it is either type (2,5) or type (3,4,5). The spectrum contains s ≤ 4 elements and depending on s we have the following possibilities: In this case T(A) is always (3,4,5).
In this case T(A) is always (3,4,5). Here , , , are different elements of the spectrum.

Proof
We know that the spectrum has at most four elements. We start with the case where there are no derivations in the subalgebra conditions. Either we have two clusters and get the only case with s = 4 or we have only one cluster of size 3 and get the second case with s = 3.
If some -derivation is used then we can suppose that it was added to a subalgebra of codimension one. If was not in the spectrum of this codimension one subalgebra, then is a trivial derivation f → f � ( ) and we get either the second case with s = 2 (with = ) or the first case with s = 3.
At last if belongs to the spectrum we can WLOG suppose that = and use that we know all -derivations. We get cases with s = 1, 2.
It is easy to check that (2,5) and (3,4,5) are the only choices for the numerical semigroup of degrees. To see which choice is valid we only need to check if the element of degree 2 in the SAGBI basis satisfies the added condition. If so we get type (2,5), otherwise type (3,4,5). Alternatively we can use Theorem 23 which tells us exactly when T(A) = (2,5) . ◻

Subalgebras of codimension three
In codimension three we can use the same approach and get a classification, but the situation is more complicated. The spectrum has at most 6 elements. We will show only how it looks for a single element in the spectrum (the complete classification can be found in [10]).

Theorem 28
If an algebra A of codimension three has a spectrum consisting of a single element then A is one of the following algebras For a = b = 0, c ≠ 0 the type is (3,4) and a SAGBI basis is with a ≠ 0. If b ≠ 0 then T(A) = (4, 5, 6, 7) and a SAGBI basis is: and we are done. The arbitrary derivation is a linear combination of f (x) → f �� ( ) and f (x) → f (5) ( ) − 10af (4) ( ) and we get the case 2 if we simply substitute 0 by back.
When we get a description there is a straightforward way described above to get a SAGBI basis: we know the possible degrees and need only to search for elements of the degrees generating the semigroup that satisfy the subalgebra conditions. ◻

About the characteristic polynomial A (x) when A has more than two generators
We would like to generalise Theorem 16 to arbitrary subalgebras. For this we need to define the characteristic polynomial for an arbitrary subalgebra. Let us look at the case where A has more than two generators. It is not evident how to extend the definition. The resultant is defined only for pairs of polynomials.
A naive attempt is to use the gcd of all g i ,g j where g i generate A. Let us first look at an example.

Example 7
Let p(x) = x 12 + 3x 6 , q(x) = x 15 and r(x) = x 10 and A = ⟨p(x), q(x), r(x)⟩ the subalgebra they generate. We can form the characteristic polynomial of any pair of generators. If we look at the pair p and q for example, it is obvious that they both belong to [x 3 ] . Hence their characteristic polynomial is zero by Theorem 15. In the same way the other two pairs of generators have zero as characteristic polynomial. In contrast, if we form P and Q as before and additionally R(x, y) = (r(x) − r(y))∕(x − y) , then P(x, y) = Q(x, y) = R(x, y) = 0 has only a finite set of solutions. In particular the possible x-values are the 24 solutions of x 24 + 6x 18 + 26x 12 + 81x 6 + 81 and x = 0 . (This can be obtained by solving the system in for example Maple.) The above example shows that looking at pairs of generators of the algebra is not enough to define the characteristic polynomial in a suitable way. Inspired by Theorem 8 we instead make the following definition.

Definition 4 Let
A be a subalgebra of finite codimension. We define its characteristic polynomial A (x) as the gcd of all p,q (x) where p and q are monic polynomials in A with relatively prime degrees.
Theorem 8 assures that the set of zeroes of A (x) equals the spectrum.
− 10a(fg) (4) (0) + (fg) (5) (0) − −10af (4) (0) + f (5) (4) (0) + g (5)  Note that D(x) depends on the generators and even on which generator is chosen as p 1 . For example if we take p 1 = r in the above example we get D(x) = x 54 b(x) . As long as we are looking for the zeroes this is not a problem, but we cannot use D as our characteristic polynomial as it is not well defined for a given subalgebra A.
But we can define characteristic polynomial as a gcd of all such D(x) to have an invariant definition. In our example it still be x 50 b(x).
We conclude by using D to find descriptions of some subalgebras on several generators.

Example 10 Let
In this case we get D(x) = x 2 (x − 2)(x + 2)(x − 1) 2 (x + 1) 2 . Thus we get that Sp(A) = {0, 1, −1, 2, −2} . Since A has generators of degrees 4, 5, 6, 7 we must again have codimension at most three. If we factor the generators it is easy to see that Finally we want to consider quite different approach to generalise the notion of characteristic polynomial for to an arbitrary subalgebra A. It is based on the observation that the characteristic polynomial belongs to the subalgebra itself. According to Theorem 19 there exists a monic polynomial p(x) such that x k p(x) ∈ A for any k ≥ 0. If we choose such polynomial of minimal degree it will be unique. (Otherwise the difference between two such polynomials would provide a polynomial of lower degree.) Let us take call such a minimal polynomial M(x).
Note that any ∈ Sp(A) is a root for M(x). This is immediate from the definition of spectrum applied to the elements M(x) and xM(x) of A. Obviously its degree is greater than the Frobenius number of the the corresponding numerical semigroup S of degrees. As we see in the last example it can be larger than the Frobenius number plus one, but we suspect it is not greater than twice the codimension. We have M(x) = x 30 b(x) in Example 8, so here the degree is exactly twice the codimension, but it is smaller than the degree of the characteristic polynomial obtained using our previous definition. It seems reasonable to think of M(x) as a kind of minimal polynomial for A, but one could also take M(x) to be the characteristic polynomial of A.

Single element in the spectrum: derivations
To understand how derivations are formed, we will study a special concrete case, algebras A with a single element in the spectrum.

3!
are two -derivations over A. Now one may ask: what -derivations exist over the kernel of some linear combination D 3 − cD 2 ? Consider the following list of the maps created with the help of Maple: Here D k is the map D k ∶ p → p (k) ( ) k! and c is a constant. We know that the first map is an -derivation. But what is more interesting is that if the first k maps defines a subalgebra inside A (as the intersection C of their kernels with A), then the next map will be a derivation over C.
For any given map in the list above, let C i be the coefficient of c k D i . I.e in the fourth map we have C 7 = 1, C 6 = −3, C 4 = −3 . Then for every single map, the following relations all hold among the C i of that map: We use parenthesised superscripts to index a particular map above. We index the maps by the highest order among the derivatives in it. Note that this means that the map with index n is in row n+1 2 in the above list. For example C (7) 6 = −3 . The following theorem states that the properties which we have observed (but not proved) the C (n) k to exhibit, uniquely determine a set of integers.

Conjecture 2
If L 1 (f ) = L 3 (f ) = ⋯ = L n−2 (f ) = 0 for each f ∈ A then L n is an -derivation in A.

Further development
Here we want to discuss some possible ways to generalise the obtained results. We have several restrictions. Are they all necessary? First of all we can consider subalgebras of infinite codimension. Then we need infinitely many conditions, so spectra can be infinite as well. But there are many interesting questions here.
Next we have the restrictions on the field. Characteristic zero seems to be important. In positive characteristic we encounter problems already in the case of monomial algebras as some derivatives vanish due to the characteristic regardless of what algebra we want to describe. But we can probably work with the divided powers.
The demand that the field is algebraically closed is probably less restrictive, at least if we allow the spectral elements to belong to the algebraic closure of the field. An interesting question related to this is to understand when the spectrum of a subalgebra over the field of complex numbers consists of real elements. It would also be interesting to investigate methods for constructing a SAGBI in this case. The main tool -the existence of a subalgebra B of codimension one less is absent. Though in the real case, we can find a subalgebra of codimension two less.
Perhaps, the most interesting generalization is to allow more than one variable. Here we need to use partial derivatives and for example the monomial subalgebras exhibit a similar description as in the univariate case. Thus there is a realistic hope for the theory to be extendable to several variables. One problem is that it is not clear that the spectrum cannot contain ghost elements if we increase the number of variables.
The main tool-containment in a subalgebra B still works but now we need (in the case of two variables) to speak about ( , )-derivations. The SAGBI bases seem to be constructed in a similar way and therefore should still be finite. But there are many differences. First of all f ( , ) = 0 does not give us a factor in f(x, y) which is a fact that we have relied substantially on in the one-dimensional case. Therefore we have no direct analogs for the proofs of theorems corresponding to Theorems 18, 19, 20. It would be interesting to know if they are still valid.
Another difference is that there exists proper subalgebras in [x, y] with empty spectrum. An example inspired by [11] is the subalgebra A = ⟨x, xy, xy 2 − y⟩.
If we assume that for all f ∈ A and apply this to the generators we find that = , = and 2 − = 2 − . If ≠ then = = 0 . Now this in turn implies that = . We conclude that ( , ) = ( , ) so the pair ( , ) was not in the spectrum of A. Similarly applied to x gives a = 0. Thus b ≠ 0 and application to xy gives = 0. But then To check that it is a proper subalgebra suppose that If we put y = 1 x here then we obtain 1 x = F(x, 1, 0) -a contradiction. In fact no y k belongs to A and we have, as expected, infinite codimension while [x, y] is the only subalgebra of finite codimension that contains A. But it is impossible to construct similar examples with finite codimension or in the one-variable case.
An interesting question is to find a homological interpretation of our results. Some kind of homological algebra should lie under the surface here.
The characteristic polynomial is especially interesting. What is the most natural way to define it? Can it be introduced for several variables? Can it be interpreted as the characteristic polynomial of some operator on V 2 or V × V * , where V = [x]∕A?
There are also fundamental open questions regarding the size of the spectrum. Is it an inner property of subalgebra? As ⟨x 2 ⟩ has an infinite spectrum, the size of the spectrum probably depends on the embedding of the subalgebra in [x]. But maybe this is not the case if we restrict ourselves by finite codimension only.
There are potential applications to important mathematical problems. We believe that the spectrum will prove to be a useful tool when comparing subalgebras.
We also hope to find some applications in cryptography because we have two essentially different ways to describe subalgebras.