Structure theorems for linear and non-linear differential operators admitting invariant polynomial subspaces

In this paper we derive structure theorems that characterize the spaces of linear and non-linear differential operators that preserve finite dimensional subspaces generated by polynomials in one or several variables. By means of the useful concept of deficiency, we can write explicit basis for these spaces of differential operators. In the case of linear operators, these results apply to the theory of quasi-exact solvability in quantum mechanics, specially in the multivariate case where the Lie algebraic approach is harder to apply. In the case of non-linear operators, the structure theorems in this paper can be applied to the method of finding special solutions of non-linear evolution equations by nonlinear separation of variables.


1.
Introduction. It is a fact that the Schrödinger operators whose point spectrum, or at least part of it, can be computed algebraically are often related to differential operators admitting invariant spaces of polynomials. Lie algebras have played a unifying role in this area, because many of these polynomial spaces turn out to be irreducible modules for a faithful representation of a finite-dimensional Lie algebra by means of first-order differential operators. The classical theory of quasi-exactly solvable potentials has thus been built on the assumption that the exactly solvable Schrödinger operator under study should be expressible as a quadratic element in the universal enveloping algebra of a finite-dimensional Lie algebra of first-order differential operators, admitting an explicitly computable invariant subspace of square-integrable functions, or a complete infinite flag thereof [1,2,3,4]. 1. In the case of polynomial subspaces in one variable, the Lie algebraic approach can only be applied to find the differential operators that leave the polynomial space P n = 1, z, z 2 , . . . , z n invariant, and it cannot characterize the set of differential operators that map P n into P m ⊂ P n with m < n. This simple problem has motivated the important notion of deficiency used throughout this paper, and applications of it can be found in the construction of solvable classical many-body problems by considering the motion of zeros of polynomials whose coefficients evolve in a controlled manner [5]. 2. Other subspaces generated by polynomials exist which are not isomorphic to P n . The Lie algebraic approach cannot be applied in these cases, as was already noted by Post and Turbiner, who characterized the spaces of linear differential operators which preserve polynomial subspaces in one variable generated by monomials. In their work [6] they solved this problem with no reference to Lie algebras. The case of a general space spanned by polynomials -referred to as the generalized Bochner problem in [6]-remains still open. Somewhat surprisingly this direct approach has not been pursued until very recently, where it has been shown that the class of quasi-exactly solvable potentials is larger than the Lie-algebraic class [7], and that even in Liealgebraic potentials other non-sl 2 algebraizations exist which allow one to obtain more levels from the energy spectrum of the Hamiltonian [8]. The existence of differential operators that preserve a general polynomial space and cannot be expressed as quadratic combinations of the generators of sl 2 is not in contradiction with Burnside's theorem, since a general polynomial space is not the carrier space for an irreducible representation of sl 2 . 3. In the case of multi-variable polynomial subspaces, the problem of characterizing the set of linear differential operators that leave these spaces invariant becomes untractable in the Lie algebraic approach. The reason is that, contrary to the single variable case where essentially sl 2 is the only algebra of first order differential operators with finite dimensional representations, in more variables many more algebras exist. But moreover, the characterization of second order operators as quadratic combinations of the generators of these algebras requires an extensive analysis of the syzygies corresponding to the primitive ideals associated to the irreducible representations [9]. These problems are entirely bypassed in the direct approach, as shown in Section 3 of this paper, where a simple characterization is given for the set of linear differential operators of any given order r that leave the simplicial module Our results coincide with the formulas for the special case N = 2 and r = 2, derived in [9] using the Lie algebraic method.
It has now become clear that the connection to Lie algebras is not an essential feature of exact or partial solvability. Our goal in the first two Sections of this paper is to present a direct method to characterize linear differential operators with invariant polynomial subspaces, which is simpler and more powerful than the Lie DIFFERENTIAL OPERATORS WITH INVARIANT POLYNOMIAL SUBSPACES 87 algebraic approach. We restrict ourselves in this paper to the simplest case of polynomial subspaces, namely the simplicial modules P n ; the case of general polynomial subspaces shall be treated elsewhere. For any number of variables N , we provide an explicit basis for the space of linear differential operators of any order r that map P n into P m ⊆ P n with m ≤ n. It should be stressed that although these results allow the construction of many differential operators with invariant finite dimensional polynomial subspaces, in general it is not known whether a transformations exists that puts the operator in Schrödinger form. Therefore the results in Sections 2 and 3 are only a first necessary step in the theory of higher dimensional quasi-exact solvability. A general theory would need to face the difficulties of the equivalence problem [10]. Despite this fact, it is worth mentioning that a few examples of partially solvable multi-dimensional Hamiltonians exist [11,12,13], which are mostly extensions of the Calogero-Sutherland class.
This paper also addresses the study of nonlinear differential operators with polynomial nonlinearities which possess invariant polynomial subspaces. The motivation for this study is twofold: 1. In [14], the important concept of operator duals is introduced. Given a finite dimensional space of functions F n = span{f 1 , . . . , f n } which are required to satisfy certain regularity conditions, the operator duals are linear differential operators defined by the relations These operators are used in the reduction of non-linear evolution equations to dynamical systems by a method of non-linear separation of variables. In [14] the existence of these differential operators is proved together with results on the regularity of the coefficients. The proof is constructive, and therefore given any space F n whose basis elements satisfy the required regularity conditions, non-linear evolution equations can be written which have solutions in the space F n ; i.e., special solutions exist of the form u(t, where the coefficients c i (t) satisfy a system of coupled non-linear ODEs. However, the order of the operator duals is precisely the dimension of the space F n . The price to pay in order to have invariant spaces of high dimension is that the order of the resulting evolution equations grows with the dimension of the space. For dimensions n larger than six, this reduces the applicative interest of the resulting equations. The motivation coming from [14] is to construct operators that generalize the operator duals, so that the order of the resulting equation and the dimension of the invariant subspace can be independently chosen. In the case of polynomial subspaces P n , the generalization of the operator duals to arbitrary order r ≤ n forms the so-called maximal deficiency operators introduced in Section 2. 2. The second motivation comes also from the results on non-linear separation of variables by King, Galaktionov and Svirshchevskii [19,15,16,17,18]. In these papers, special interest is given to translation-invariant evolution equations with quadratic non-linearities which admit solutions via non-linear separation of variables. From a physical context, applications are found in nonlinear diffusion and thin film equations. In this paper, we extend these results by providing a comprehensive structure theory for autonomous nonlinear operators that preserve a polynomial space P n .

GÓMEZ-ULLATE, KAMRAN AND MILSON
Our paper is organized as follows. In Section 2, we present our direct approach in the case of linear differential operators in one variable. Besides the order of a differential operator, we introduce two key invariants that can be freely specified, which are the degree and the deficiency of the operator relative to a polynomial space P n . The order, degree and deficiency are shown to specify the operator uniquely up to scaling by a constant. The operators of maximal deficiency generalize the operator duals of [14] to any order lower than n. We give an explicit basis for the space of operators of given order and deficiency. Section 3 is concerned with linear differential operators in N variables, where all the results of Section 2 are shown to extend to the case of simplicial modules, i.e., multivariate polynomials of total degree bounded by a given integer. Section 4 is concerned with non-linear operators preserving polynomial modules, where we give an explicit decomposition theorem for the most general non-linear operator with polynomial coefficients preserving a simplicial module. Section 5 studies the deficiency concept for non-linear operators with polynomial nonlinearities. Section 6 concentrates on non-linear operators that are translation invariant (autonomous), deriving also structure theorems for this class. In Section 7 the application of these results to non-linear separation of variables is discussed, while some explicit formulas are given for quadratically non-linear autonomous operators in Appendix A.
2. Linear operators in one variable. In this section we consider the class of scalar linear differential operators on the real line, with polynomial coefficients. We are interested in the subclass of operators which have a definite order, degree and deficiency. These quantities are defined as follows. The order of is as usual the largest r for which the coefficient a r (x) is not identically zero. We say that L is of degree d ∈ Z if for all j ∈ N, there exists c j = 0 ∈ R not all zero such that In order to define the deficiency, we fix n ∈ N and consider the vector space P n = span{1, x, . . . , x n } of polynomials in x of degree less than or equal to n. We say that L has deficiency m ∈ Z relative to P n if LP n ⊂ P n−m , but LP n ⊂ P n−m−1 .
Let L (n) r,m denote the set of linear differential operators with polynomial coefficients, of order less than or equal to r and of deficiency greater than or equal to m relative to P n . Again, we emphasize that the notion of operator deficiency only makes sense relative to a particular n. Most of our discussion will be carried out with the assumption that the n in P n has been fixed. As such, we will often omit the n in our terminology and notation, simply speak of the deficiency of an operator, and write L r,m instead of L Proof. Given linear operators L, L ′ ∈ L r,m , the order of any linear combination of L and L ′ is less than or equal to r. Similarly, the deficiency of a linear combination is greater than or equal to m. Proposition 2. The deficiency of a non-zero linear operator cannot exceed its order.
Proof. Suppose that L ∈ L r,m is a linear operator such that m > r. The operator D n−m+1 L annihilates P n , but it has order less than n + 1. This is impossible.
Trivial examples of operators with given order, degree and deficiency are given by the operator D i , which has order i, degree −i and deficiency i, and the multiplication operator x j , which has order zero, degree j and deficiency −j. These operators do not depend on the degree n of the polynomial space P n . A more significant example, which depends explicitly on n, r, m, d with 0 ≤ m ≤ r ≤ n and −m ≤ d ≤ r − m, is the operator where we have introduced the Pochhammer operator A basic result is the following proposition.
Proposition 3. The operator L rmd has order r, degree d and deficiency m.
Proof. Since the k-fold composition appearing in the right-hand-side of (1) annihilates the monomials x n , . . . , x n−k+1 , the operator in (1) has order k, degree zero and deficiency k. It follows that the operator ( n − j − xD ) k D j has order r = k + j, deficiency m = k + j and degree d = −j. By left-multiplying by a monomial of x, we raise the degree and lower the deficiency so that L rmd has order r, deficiency m and degree d.
Following Proposition 2, it is helpful to refer to an operator whose deficiency equals its order as a maximal deficiency operator. To this end, we will use the symbol to denote the operators of maximal deficiency. The normalization constant of 1 (r−j)! will be useful in later formulas. These operators have a number of interesting properties and play a key role in our theory.
Proposition 4. Up to multiplication by a non-zero real constant, the operator K rj is the unique r th order maximal deficiency operator with polynomial coefficients and Proof. Let L be an r th order maximal deficiency operator with polynomial coefficients and degree d. Since L has polynomial coefficients, d ≥ −r. Also, L maps x k to a multiple of x k+d . Hence, However, a non-zero r th -order operator can annihilate at most r monomials. Thus, we have established that 0 ≤ j ≤ r, where j = −d.
The leading order term of both L and K rj is a multiple of x r−j D r . Hence, there exists an a ∈ R such that the order of L − aK rj is less than r. However, the 90 GÓMEZ-ULLATE, KAMRAN AND MILSON deficiency of L − aK rj is greater than or equal to r. Therefore, by Proposition 2, we have L = aK rj .
Proof. The operators ( n − j − xD ) r−j D j and D j ( n − xD ) r−j both have order r, deficiency r and degree −j. By Proposition 4 they differ by a scalar multiple. By comparing the coefficients of the leading order, we see that these two operators are actually equal.
Proposition 6. For a fixed r, the operators K rj are recursively defined by Proof. Setting k = r − j, we have, by Proposition 5, We can also expand the maximal deficiency operators into an operator sum.
Proposition 7. We have Proof. LetK rj denote the right hand side of (2). A direct calculation shows that Hence, by Proposition 6, K rj andK rj have the same recursive definition. Therefore, the two operators are equal.
Proposition 8. We have dim L r,r = r+1. Indeed, the operators K rj , j = 0, 1, . . . , r form a basis for the vector space L r,r .
Proof. Let L ∈ L r,r be given. Decomposing L into operator monomials of homogeneous degree, we have, by Proposition 2 and the fact that P n is generated by monomials, Since the K rj are linearly independent, we conclude that dim L r,r = r + 1.
The following corollary follows immediately.
In particular, the set of operators of order r or less that map P n to itself is of dimension given by (r + 1) 2 = dim gl(r + 1, R).
We conclude this section with the following simple example. Example 1. A basis for the vector space L 2,2 of operators of order two and deficiency two is given by 3. Linear operators in several variables. The results of the preceding section extend readily to the case of linear operators in N variables (x 1 , . . . , x N ). We consider the vector space , of polynomials of degree n in N variables, and of dimension N +n n . We shall use the standard multi-index notation, whereby given a multi-index I = (i 1 . . . i N ) ∈ N N , we let The notions of order, degree and deficiency are defined similarly to the singlevariable case. Thus, an operator In order to define the deficiency, we fix again n ∈ N and say that L has deficiency m ∈ Z relative to P N n if LP N n ⊂ P N n−m , but LP N n ⊂ P N n−m−1 . As in the single variable case, it is easy to see that the deficiency of an operator cannot exceed its order. For example, the operator D I has order |I|, degree −|I|, and deficiency |I|, and the operator x I has order zero, degree |I|, and deficiency −|I|. Again, a more significant example is obtained by introducing the Euler operator and considering the operator

GÓMEZ-ULLATE, KAMRAN AND MILSON
This operator has order |J| + k, degree |I| − |J| and deficiency |J| + k − |I|. Again, if we let L r,m denote the vector space of linear differential operators in N variables (x 1 , . . . , x N ), with polynomial coefficients, and of order r and deficiency m, then we have dim L r,m = N + r − m r − m N + r r or, equivalently, L r,m = P r−m ⊗ L r,r . This formula is in agreement with the result obtained in [9] for operators in two variables, in which it was proved that dim L r,0 = 2 + r r 2 .
The proof given in [9] was less direct and required an analysis of the syzygies defined by the primitive ideals associated to the irreducible representations of sl(3, R).
It is easy to see that in contrast with the single variable case, the order, degree and deficiency are not sufficient to characterize an operator uniquely up to a nonzero factor. We have the following proposition.
Proposition 10. A basis for the vector space of linear differential operators with polynomial coefficients, of order r, degree d and deficiency m, and in N variables is given by This vector space is thus of dimension 4. Non-linear operators. Our objective in this section is to show that the results of the two preceding sections can be applied to prove a structure theorem for a class of non-linear differential operators admitting invariant polynomial subspaces.
We shall see that these results complement the structure theorems for operators preserving simplicial modules, which were proved in [14].
We will deal only with polynomial non-linearities, and to this end we introduce the following product notation for the product of two linear operators L 1 and L 2 acting on a function f (x): There is an obvious isomorphism ϕ between non-linear operators and differential functions on jet space. To that end, let denote the bundle of r-jets of smooth maps from R to R. The r-th prolongation of a smooth, real-valued function f (x) is a section of J r ; namely, pr r (f ) = (f, Df, D 2 f, . . . , D r f ).
Thus, the action of an operator T on a function of x is the same thing as the composition of a differential function ϕ(T ) of the jet variables with the prolongation: We introduce the standard jet coordinates x, u 0 = u, u 1 = u x , u 2 = u xx , . . . , u r on J r so that D j [f ] = u j • pr r (f ); i.e., ϕ(D j ) = u j . The isomorphism ϕ transforms naturally the product of operators ⊙ into the product of functions: ϕ(L 1 ⊙ L 2 ) = ϕ(L 1 ) ϕ(L 2 ). Since a differential function on jet space ϕ(T ) is clearly identified with a non-linear operator T , we shall adopt the convention of denoting both of them with the symbol T , as long as no ambiguity can arise.
Henceforth, we fix n. By Propositions 8 and 9, all linear operators of order r are expressed uniquely as polynomial linear combinations of the maximal deficiency operators K rj . Thus, a linear operator K rj maps P n to P n−r ; a quadratically nonlinear K ri ⊙ K rj maps P n to P 2(n−r) ; a cubically nonlinear K ri ⊙ K rj ⊙ K rk maps P n to P 3(n−r) , etc. This implies the following proposition.
Proposition 11. Every operator ( linear or non-linear ) of order r can be uniquely expressed as The operator in question will map P n to P n if and only if p ∈ P n , p i ∈ P r , p ij ∈ P 2r−n , p ijk ∈ P 3r−2n , . . .
The above proposition has the following obvious consequence. Corollary 1. An operator of order r < n 2 and mapping P n to P n is necessarily linear. If r < 2n 3 , then the operator will have at most quadratic non-linearities. Conversely, the above proposition can be used to bound the degree of any polynomial space which can be left invariant by a non-linear operator. We have, for example, the following corollary.
Corollary 2. If a second-order (r = 2) non-linear operator preserves a polynomial space P n , then necessarily n ≤ 4.
The extension of these results to multivariate differential operators acting on simplicial polynomial modules is straightforward. We are interested in writing down all non-linear differential operators of order r that preserve We recall that for fixed order r, the maximum deficiency that an operator can attain is r, which is achieved by any of the following maximum deficiency operators: The extension of Proposition 11 to the case of multivariate differential operators and polynomial modules is straightforward by simply substituting the simple indices i, j into multi-indices I, J, each of which can assume dim L r,r = N +r r different values. Similarly, the two Corollaries hold verbatim in the multivariate case. Example 2. Write down all second order operators with quadratic non-linearities that map P 4 into itself. A direct application of Proposition 11 allows us the following:

GÓMEZ-ULLATE, KAMRAN AND MILSON
where p ij = p ji are constants and The resulting quadratic combination is the following six parameter family of operators: If we are interested in finding only autonomous non-linear equations, then we need to consider the translation-invariant subfamily of the above operators, i.e., operators where the variable x does not appear in the coefficients. Imposing this condition leads to p 10 = p 12 = p 22 = 0; p 11 + p 20 = 0, so that the only autonomous second order non-linear operators which preserve the space P 4 are: The first of these two operators is easily seen to map P 4 into itself. The second operator is more interesting. Each of the non-linear terms u 2 x and u xx u transform a 4 th -degree polynomial into a 6 th -degree polynomial. However, the linear combination 4u xx u − 3u 2 x cancels the coefficients of degree 6 and 5, and hence it defines a map from P 4 to itself (see also [15] and [19]).
We investigate further into the analysis of autonomous non-linear operators with invariant polynomial subspaces in the following section.

5.
The algebra of polynomially non-linear operators. In this section we continue the study of non-linear operators that are polynomial in the function and its first n derivatives. First, let us fix some notation. Let T = R[x, u 0 , . . . , u n ] denote the commutative algebra of non-linear operators that can be expressed as polynomials over R in x and the derivatives u j . Multiplication in this algebra is by pointwise multiplication rather than operator composition. We grade this algebra by total degree in the u j variables: We will refer to the integer ℓ as an operator's degree of non-linearity. Thus, T 1 is the vector space of linear operators, T 2 is the vector space of quadratically nonlinear operators, etc. The vector space T 0 is the space of constant operators. For example, the operator x maps all of P n to x. Thus T 0 is a subalgebra of T , which is isomorphic to the polynomial algebra in the variable x. All the other T ℓ , ℓ ≥ 1, are merely subspaces of T , not subalgebras.
We further grade each T ℓ according to the following monomial weighting scheme: Thus where is the subspace generated by monomials having weight k. We will refer to the integer n − k as the monomial deficiency. If M = x j u i1 u i2 . . . u i ℓ is an operator monomial with then, in accord with the above-introduced meaning of deficiency, M maps P n into P k but not into P k−1 .
In other words, every operator T ∈ T admits the unique decomposition where and where the sum is taken over finitely many values of ℓ and k. For generic values of the coefficients C ji1...i ℓ , the operator T ℓ,k has deficiency n − k. However, for certain very specific values of the coefficients, the actual deficiency is greater than the monomial deficiency. This is so because in the linear combination (6), there might occur some cancellations in the terms of highest degree. We see for instance in Example 2 that relative to P 4 , the operator has monomial deficiency 4 − k = 4 − 8 + 2 = −2. However, the actual deficiency of this operator is zero. Next, we describe generators for T that will allow us to precisely determine the deficiency of an operator. Following Proposition 7, let us re-introduce the linear, n th -order operators of maximal deficiency: Thus, v n−2 = u n−2 − xu n−1 + 1 2 x 2 u n−2 , v n−3 = u n−3 − xu n−2 + 1 2 x 2 u n−3 − 1 6 x 3 u n−4 . . . .

GÓMEZ-ULLATE, KAMRAN AND MILSON
These operators transform all elements of P n into a constant. In other words, the v j are the operator duals [14] to the monomial basis of P n : More generally, let us writẽ In this way, v j =ũ j (−x).
Proposition 12. We haveũ Hence, u j →ũ j (t) defines a 1-parameter transformation group of T . The desired result follows immediately. Now, we can invert the relations (7) and express the u j in terms of the v j .
Proposition 13 shows that x, v 0 , . . . , v n freely generate the algebra T . The relations (7) and (12) are homogeneous relative to the weights (3). Hence, setting we recover the grading by monomial deficiency relative to this basis. We now deepen the grading by defining Proof. Since the relations (7), (12) are homogeneous relative to the weighting scheme (3), (13), the elements of T ℓ,k,m have weight k and, hence, monomial deficiency n − k. Since the operators v j map all polynomial to constants, an operator Q(x, v 0 , . . . , v n ) ∈ T , having m as its x-degree, maps P n to P m but not to P m−1 . Therefore such an operator has deficiency n − m.
The key application of the above grading has to do with the decomposition of an operator according to monomial deficiency. Our analysis would be greatly simplified if we could be certain that the decomposition of a non-linear operator T according to monomial deficiency respects the actual deficiency. In other words, when considering operators of a fixed deficiency, no generality is lost by considering operators that are homogeneous in degree of non-linearity and monomial deficiency.
Corollary 3. Let T be a non-linear operator whose deficiency is n−m or more; i.e., T maps P n into P m . Let T ℓ,k be the summands of the decomposition of T according to degree of non-linearity and monomial deficiency, as per (5), (6). Then, each T ℓ,k also maps P n into P m .

Autonomous, non-linear operators. Our main focus in this section is the subalgebra
of translation-invariant non-linear operators. The subalgebra inherits the bi-grading relative to degree of non-linearity and monomial deficiency, with Our key result in this section is the characterization of the deficiency of autonomous operators. In other words, we describe A ∩ T ℓ,k,m . The obvious approach to constructing non-linear operators of deficiency m would be to write a generic polynomial p(x) ∈ P n with indeterminate coefficients, act on it by (5), where the degree of the possible non-linearities is bounded by Proposition 11, and impose that the coefficients of all the terms in x j for j > n − m vanish. However, based on the useful concept of deficiency, we choose here to adopt a somewhat different approach.
We seek operators that are translation-invariant and have maximum deficiency. To this end, we define ξ = u n−1 u n ; and note that I 0 = u n , I 1 = 0.
The jet space function ξ is only defined on open neighborhood u n = 0 of J n . Thus, the operators I j are defined only for elements of P × n = P n \ P n−1 , the set of polynomials of degree exactly equal to n. We can still speak of the deficiency of such operator, but this has to be understood in terms of P × n rather than P n . Proposition 15. The non-linear operators I 2 , I 3 . . . , I n are translation-invariant and have maximum deficiency n. In other words, these autonomous, nonlinear operators transform every n th degree polynomial into a constant.
Proof. By definition, v j =ũ j (−x), I n−j =ũ j (−ξ). Also, Hence, Proposition 12 implies that in addition to (15), we also have Hence, the operators I j are polynomials of the v j divided by a certain power of v n = u n . Therefore, these operators are both translation invariant and of maximal deficiency.
We can also invert the relations (15) and express the u j in terms of the I j .
Thus, relations (15), (18) tell us that the operators I 0 , ξ, I 2 . . . , I n also generate the algebra of autonomous operators. These relations are homogeneous with respect to the monomial weights defined in (3). Hence, setting wt(I j ) = j, wt(ξ) = 1, we recover the grading by monomial deficiency relative to this basis. Unfortunately, the operators I j , ξ are not polynomials in the u j , and hence they are not elements of A. Let us therefore consider the larger algebrâ A = R[I 0 , ξ, I 2 , I 3 , . . . , I n ], generated by ξ and the autonomous operators of maximal deficiency. Thanks to (18) we know that A ⊂Â, but this inclusion is strict.
Proof. This follows from (16), (17), and the fact that the operators v j have maximal deficiency.
In other words, the deficiency of an autonomous operator is n minus the ξ-degree of the polynomial that expresses that operator relative to the ξ, I j basis. We are left with the question of the nature of the inclusion of A inÂ, i.e., the question of which polynomials in ξ, I j define true polynomial operators.
Theorem 1. Let T = P (x, u 0 , . . . , u n ) ∈ T be a non-linear operator, and let T = Q(x, v 0 , . . . , v n ) be the expression of this operator relative to the non-autonomous generators x, v j . Then T is autonomous; i.e., P is independent of the variable x if and only if T = Q(ξ, I n , . . . , I 2 , 0, I 0 ). In this case, the deficiency of T is equal to n minus the ξ-degree of the polynomial Q(ξ, I n , . . . , I 0 ).

Proof. By Propositions 13 and 16 we have
where, as we noted before, I 1 = 0, and I 0 = v n = u n .
Example 3. Let us recast the analysis begun in Example 4 in terms of the above operator bases. Relations (12) and (18) take the form Our goal is to write down all autonomous operators with quadratic non-linearity and zero deficiency, i.e., operators that map P 4 into itself. By Proposition 17 and the above relations, we are obliged to consider polynomials that are quadratic in I 0 , I 1 , I 2 , I 4 and that have degree 4 or less in the ξ variable. The question is, which operators of such form are quadratic in u 0 , u 1 , u 2 , u 3 , u 4 ? By Corollary 3, no generality is lost by considering operators of a fixed monomial deficiency. Evidently, if the monomial deficiency is 0 or more, then the operator will preserve P 4 . Hence we must consider operators having monomial deficiency −4, −3, −2, −1, which corresponds to k = 8, 7, 6, 5, respectively. The most general operator having k = 8 is a multiple of the monomial (u 0 ) 2 . Such an operator will have an actual deficiency of −4 as well. Similar reasoning holds for operators with k = 7; these are multiples of u 0 u 1 . Let us consider the case k = 6. The ansatz is now  Hence, such an operator preserves P 4 if and only if is a multiple of 4u 0 u 2 − 3u 2 1 , as has already been proven by other methods.
Finally, let us consider the case k = 5. Now, the ansatz is Hence, such an operator preserves P 4 if and only if is a multiple of 2u 0 u 3 − u 1 u 2 . Indeed, because I 1 = 0, the above calculation shows that this operator has deficiency 1; i.e., it maps P 4 into P 3 .
Autonomous operators with quadratic non-linearities. We restrict our attention from here on to T 2 , the vector space of operators with homogeneously quadratic non-linearity. We present a complete characterization of such operators in terms of deficiency, thereby extending the results of Svirshchevskii [19] and Galaktionov [15]. Our analysis can be extended to operators with higher non-linearities, but this shall be treated elsewhere. Following (4), we let be the linear space of quadratic autonomous non-linear operators up to order n, graded according to monomial deficiency. Thus, where ⌊·⌋ denotes the floor function. Each of the elements of the above basis of Q k has a different order r, the minimum and maximum orders for each k being r (k) min = n + k 2 , r (k) max = min(n, n + k).
All of the above basis elements have the same deficiency k; i.e., they map P n into P n−k . However, Example 4 and Example 6 show that within each Q k there can be elements whose deficiency is greater than k.
Remarkably, each Q k possesses an adapted basis that further grades it according to operator order and deficiency. In this regard, for r (k) min ≤ r ≤ r (k) max , let us introduce the operators We are now ready to state the main result of this Section.
Main Theorem. The operator Q k,r has monomial deficiency k, order r, and deficiency This result is similar to that proved in Proposition 4 for linear operators: up to a scalar multiple there is only one quadratic autonomous operator with a given order r, monomial deficiency k, and deficiency m(k, r). In Appendix A we show the explicit form of these operators for n = 4, 5 and 6.
We will prove the theorem by finding a generating function for the operators Q k,r . This requires some new notation. For a bivariate formal series, to be the series formed from terms where the two variables have equal exponents. We will also adopt the convention that Proof of Main Theorem . We extend the definition (20) of Q k,r to all 0 ≤ r ≤ n by setting Q k,r = min(n,n+k) i=max(0,k) We can now form a generating function for Q k,r as follows: where s = 2n − i − j − p − q and we used relation (19) to replace the u j with the I j , the autonomous generators of maximal deficiency defined by (15). Let us extend the definition of I j to all j ≥ 0 by setting and agreeing that u j = 0 for j < 0. However, we must note that the newly defined autonomous operators I j , j > n are no longer maximal deficiency operators.