G N ] 8 J an 2 02 1 Separately polynomial functions

It is known that if f : R2 → R is a polynomial in each variable, then f is a polynomial. We present generalizations of this fact, when R2 is replaced by G × H, where G and H are topological Abelian groups. We show, e.g., that the conclusion holds (with generalized polynomials in place of polynomials) if G is a connected Baire space andH has a dense subgroup of finite rank or, for continuous functions, if G and H are connected Baire spaces. The condition of continuity can be omitted if G and H are locally compact or complete metric spaces. We present several examples showing that the results are not far from being optimal.


Introduction
It was proved by F. W. Carroll in [2] that if f : R 2 → R is a polynomial in each variable, then f is a polynomial. Our aim is to find generalizations of this fact, when R 2 is replaced by the product of two topological Abelian groups.
On topological Abelian groups we distinguish between the class of polynomials and the wider class of generalized polynomials (see the next section for the definitions). The two classes coincide if the group contains a dense subgroup of finite rank. Now, the scalar product on the square of a Hilbert space is an example of a continuous function which is a polynomial in each variable without being a polynomial (see Example 1 below). Therefore, the appropriate problem is to find conditions on the groups G and H ensuring that whenever a function on G × H is a generalized polynomial in each variable, then it is a generalized polynomial.
We show that this is the case if G is not the union of countable many zero sets of generalized polynomials, and if H has a dense subgroup of finite rank (Theorem 4). The condition on G is satisfied, for example, if G is a connected Baire space. Note that the continuity of f is not assumed in Theorem 4.
If G and H are both connected Baire spaces, and if a continuous function on G ×H is a generalized polynomial in each variable, then it is a generalized polynomial (Corollary 12).
It is not clear if the condition of continuity can be omitted from Corollary 12 (see Question 13). The problem is that a generalized polynomial must be continuous by definition, and a separately continuous function on the product of Baire spaces can be discontinuous everywhere, as it was shown recently in [9]. In our case, however, there are some extra conditions: the spaces are also connected, and the function in question is a generalized polynomial. It is conceivable that continuity follows under these conditions. As for the biadditive case, see [3].
We show that if G and H are connected complete metric spaces, or they are connected and locally compact, then every separately generalized polynomial function on G × H is a generalized polynomial (Theorem 14).
The proof shows that the conclusion holds whenever G and H are connected Baire spaces and such that every separately continuous function on G × H has at least one point of joint continuity.

Preliminaries
Let G be a topological Abelian group. We denote the group operation by addition, and denote the unit by 0. The translation operator T h and the difference operator ∆ h are defined by We say that a continuous function f : The smallest n with this property is the degree of f , denoted by deg f . The degree of the identically zero function is −1. We denote by GP = GP G the set of generalized polynomials defined on G.
A function f : G → C is said to be a polynomial, if there are continuous additive functions a 1 , . . . , a n : G → C and there is a P ∈ C[x 1 , . . . , x n ] such that f = P (a 1 , . . . , a n ). It is well-known that every polynomial is a generalized polynomial. It is also easy to see that the linear span of the translates of a polynomial is of finite dimension. More precisely, a function is a polynomial if and only if it is a generalized polynomial, and the linear span of its translates is of finite dimension (see [8,Proposition 5]). We denote by P = P G the set of polynomials defined on G.
Let f be a complex valued function defined on X × Y . The sections f x : Y → C and f y : Let G, H be topological Abelian groups. A function f : (G × H) → C is a separately polynomial function if f x ∈ P H for every x ∈ G and f y ∈ P G for every y ∈ H. Similarly, we say that f : (G × H) → C is a separately generalized polynomial function if f x ∈ GP H for every x ∈ G and f y ∈ GP G for every y ∈ H.
In general we cannot expect that every separately polynomial function on G × H is a polynomial; not even if G = H is a Hilbert space. Example 1. Let G be the additive group of an infinite dimensional Hilbert space. Then the scalar product f (x, y) = x, y on G 2 is a separately polynomial function, since its sections are continuous additive functions. In fact, f y is a linear functional and f x is a conjugate linear functional for every x, y ∈ G. Thus the sections of f are polynomials. Now, while the scalar product is a generalized polynomial (of degree 2) on G 2 , it is not a polynomial on G 2 , because the dimension of the linear span of its translates is infinite. Indeed, let g(x) = x, x = x 2 for every x ∈ G. Then ∆ h g(x) = 2 h, x + h 2 for every h ∈ G. It is easy to see that the functions h, x (h ∈ G) generate a linear space of infinite dimension, and then the same is true for the translates of g and then for those of f as well.
Therefore, the best we can expect is that, under suitable conditions on G and H, every separately generalized polynomial function on G × H is a generalized polynomial.
We denote by r 0 (G) the torsion free rank of the group G; that is, the cardinality of a maximal independent system of elements of G of infinite order. Thus r 0 (G) = 0 if and only if G is torsion. In the sequel by the rank of a group we shall mean the torsion free rank. It is known that if G has a dense subgroup of finite rank, then the classes of polynomials and of generalized polynomials on G coincide (see [8,Theorem 9]).
The set of roots of a function f : It is easy to see that N P and N GP are proper ideals of subsets of G. Let N σ P and N σ GP denote the σ-ideals generated by N P and N GP , respectively. Note that N P ⊂ N GP and N σ P ⊂ N σ GP .
If G is discrete, then N σ P and N σ GP are not proper σ-ideals (except when G is torsion), according to the next observation.
Proof. Let a ∈ G be an element of infinite order. Then φ(na) = n (n ∈ Z) defines a homomorphism from the subgroup generated by a into Q, the additive group of the rationals. Since Q is divisible, φ can be extended to G as a homomorphism from G into Q. Let ψ be such an extension.
A simple sufficient condition for G / ∈ N σ GP is given by the next result.
Proof. It is enough to prove that every element of N GP is nowhere dense.

Main results
Our next result generalizes Carroll's theorem [2].
Theorem 4. Let G, H be topological Abelian groups, and suppose that H has a dense subgroup of finite rank.
If f : (G × H) → C is a separately generalized polynomial functions, then f is a generalized polynomial on G × H.
Remark 5. By Lemma 3, (i) of Theorem 4 can be replaced by the condition that G is a connected Baire space.

Lemma 6.
Let H be a topological Abelian group, and suppose that H has a dense subgroup of finite rank. Then, for every positive integer d, there are finitely many points x 1 , . . . , x s ∈ H and there are generalized polynomials Proof. Let GP <d denote the set of generalized polynomials f ∈ GP H of degree < d. Clearly, GP <d is a linear space over C.
Let K be a dense subgroup of H with r 0 (K) = N < ∞. Let {h 1 , . . . , h N } be a maximal set of independent elements of K of infinite order, and let L denote the subgroup of K generated by the elements First we prove that if p ∈ GP <d vanishes on A, then p = 0.
Suppose p = 0. Since p is continuous and K is dense in H, there is an x 0 ∈ K such that p(x 0 ) = 0. The maximality of the system {h 1 , . . . , h N } implies that nx 0 ∈ L with a suitable nonzero integer n. It is easy to see that there is a polynomial P ∈ C[x] such that p(mx 0 ) = P (m) for every integer m. Since P (1) = p(x 0 ) = 0, it follows that P = 0, hence P only has a finite number of roots. Thus p(mnx 0 ) = P (mn) = 0 for all but a finite number of integers m. Fix such an m. Then mnx 0 ∈ L, and thus mnx 0 = k, h with a suitable k ∈ Z N . We find that p( k, h ) = 0 for some k ∈ Z N . Let k ∈ Z N be such that p( k, h ) = 0 and k is minimal. If k ≤ [d/2], then k, h ∈ A, and we have p( k, h ) = 0 by assumption. Thus we have k > [d /2]. Then there is an ℓ = (ℓ 1 , . . . , ℓ N ) such that ℓ i ∈ {−1, 0, 1} for every i = 1, . . . , N, and k − jℓ < k for every j = 1, . . . , d. By the minimality of k we have p( k − jℓ, h ) = 0 for every j = 1, . . . , d.
which is impossible. This proves p = 0.
The set of functions V = {p| A : p ∈ GP <d } is a linear space over C. The map p → p| A is linear from GP <d onto V and, as we proved above, it is injective. Therefore, GP <d is of finite dimension. Then there are functions q 1 , . . . , q s ∈ GP <d such that q i (x i ) = 1 and q i (x j ) = 0 for every i, j = 1, . . . , s, i = j.
Let p ∈ GP <d be given. Then p− s i=1 p(x j )q j is a generalized polynomial of degree < d vanishing on X, hence on H. That is, we have p = s i=1 p(x j )q j .
Proof of Theorem 4. Let f : (G × H) → C be a separately generalized polynomial function. Put G n = {x ∈ G : deg f x < n} (n = 1, 2, . . .). Since N σ GP (G) is a proper σ-ideal in G, there is an n such that G n / ∈ N GP (G). Fix such an n.
By Lemma 3, there are points y 1 , . . . , y s ∈ H and generalized polynomials q 1 , . . . , q s ∈ GP H such that p = s i=1 p(y i ) · q i for every p ∈ GP H with deg p < n. Therefore, we have for every x ∈ G n and y ∈ H. If y ∈ H is fixed, then f (x, y)− s i=1 f (x, y i )q i (y) is a generalized polynomial on G vanishing on G n . Since G n / ∈ N GP (G), it follows that f (x, y) = s i=1 f (x, y i )q i (y) for every (x, y) ∈ G×H. By f y i ∈ GP G and q i ∈ GP H , we obtain f ∈ GP G×H .
Next we show that in Theorem 4 none of the conditions on G and H can be omitted. First we show that without condition (i) the conclusion of Theorem 4 may fail. We shall need the easy direction of the following result. Proof. Suppose f : (G × H) → C is a generalized polynomial of degree < d.
Then, for every y ∈ H, we have ∆ (x 1 ,0) . . . ∆ (x d ,0) f y = 0 for every x 1 , . . . , x d ∈ G, and thus f y is a generalized polynomial of degree < d for every y ∈ H. A similar argument shows that f x is a generalized polynomial of degree < d for every x ∈ G, proving the "only if" statement.
Now suppose that f : (G × H) → C is such that f x (x ∈ G) and f y (y ∈ H) are generalized polynomials of degree < d. Then we have for every h 1 , . . . , h d ∈ G and y ∈ H, and for every k 1 , . . . , k d ∈ H and x ∈ G. In order to prove that f is a generalized polynomials of degree < 2d, it is enough to show that By Proposition 2, N σ P (G) is not a proper σ-ideal; that is, G = ∞ n=1 A n , where A n = ∅ and A n ∈ N P (G) for every n. Let p n ∈ P G be such that p n = 0 and A n ⊂ Z pn . Then p n is not constant; that is, deg p n ≥ 1.
Let P n = p 1 · · · p n ; then P n (x) = 0 for every x ∈ n i=1 A i , and we have deg P 1 < deg P 2 < . . .. (Here we use the fact that deg pq = deg p + deg q for every p, q ∈ GP G , p, q = 0.) Note that for every x ∈ G we have P n (x) = 0 for all but a finite number of indices n.
Similarly, we find polynomials Q n ∈ P H such that deg Q 1 < deg Q 2 < . . ., and for every y ∈ H we have Q n (y) = 0 for all but a finite number of indices n.
We put f (x, y) = ∞ n=1 P n (x)Q n (y) for every x ∈ G and y ∈ H. If y ∈ H is fixed, then the sum defining f is finite, and thus f y ∈ P G . Similarly, we have f x ∈ P H for every x ∈ G.
The degrees deg f y (y ∈ H) are not bounded. Indeed, for every N, there is an y ∈ H be such that Q N (y) = 0. Then f y = M n=1 Q n (y) · P n with an M ≥ N, where the coefficients Q n (y) are nonzero if n ≤ N. Therefore, deg f y ≥ deg P N ≥ N, proving that the set {deg f y : y ∈ H} is not bounded. By Lemma 3, it follows that f is a not a generalized polynomial.
By the example above, if G and H are discrete Abelian groups of positive and finite rank, then the conclusion of Theorem 4 fails. That is, G / ∈ N σ GP (G) cannot be omitted from the conditions of Theorem 4.
Next we show that the condition on H cannot be omitted either.
Example 9. Let H be a discrete Abelian group of infinite rank. We show that if G is a topological Abelian group such that P G contains nonconstant polynomials, then there is a continuous separately polynomial function f on G × H such f is not a generalized polynomial.
Let h α (α < κ) be a maximal set of independent elements of H of infinite order, where κ ≥ ω. Let K denote the subgroup of H generated by the elements h α (α < κ). Every element of K is of the form α<κ k α h α , where k α ∈ Z for every α, and all but a finite number of the coefficients k α equal zero.
Let p ∈ P G be a nonconstant polynomial. We define f (x, y) = ∞ i=1 k i · p i (x) for every x ∈ G and y ∈ K, y = α<κ k α h α . (Note that the sum only contains a finite number of nonzero terms for every x and y.) In this way we defined f on G × K such that f x is additive on K for every x ∈ G.
If y ∈ H, then there is a nonzero integer n such that ny ∈ K. Then we define f (x, y) = 1 n · f (x, ny) for every x ∈ G. It is easy to see that f (x, y) is well-defined on G × H, and f x is additive on H for every x ∈ G. Therefore, f x is a polynomial on G for every x ∈ G.
If y ∈ H and ny ∈ K for a nonzero integer n, then f y is of the form 1 n · N i=1 k i · p i , and thus f y ∈ P G . Since f y is continuous for every y ∈ H and H is discrete, it follows that f is continuous on G × H.
Still, f is not a generalized polynomial on G × H, as the set of degrees deg f y (y ∈ H) is not bounded: if y = h i , then f y = p i , and deg p i = i · deg p ≥ i for every (i = 1, 2, . . .).
In the example above we may choose G in such a way that G / ∈ N σ GP (G) holds. (Take, e.g., G = R.) In our next example this condition holds for both G and H.
Example 10. Let E be a Banach space of infinite dimension, and let G be the additive group of E equipped with the weak topology τ of E. Then G is a connected topological Abelian group. It is well-known that every ball in E is nowhere dense w.r.t. τ , and thus G is of first category in itself.
Still, we show that G / ∈ N σ GP (G). Indeed, the original norm topology of E is stronger than τ , and makes E a connected Baire space. If a function is continuous w.r.t. τ , then it is also continuous w.r.t. the norm topology. Therefore, every polynomial p ∈ P(G) is also a polynomial on E, and thus N P (G) ⊂ N P (E) and N σ P (G) ⊂ N σ P (E). Since N σ P (E) is proper by Lemma 3, it follows that N σ P (G) is proper. The same is true for N σ GP .
Now let H be an infinite dimensional Hilbert space, and let G be the additive group of H equipped with the weak topology of H. Let f be the scalar product on H 2 . Since the linear functionals and conjugate linear functionals are continuous w.r.t. the weak topology, it follows that f is a separately polynomial function on G 2 (see Example 1).
However, f is not a generalized polynomial on G 2 , since f is not continuous. In order to prove this, it is enough to show that f (x, x) = x 2 is not continuous on H w.r.t. the weak topology. Suppose it is. Then there is a neighbourhood U of 0 such that x < 1 for every x ∈ U. By the definition of the weak topology, there are linear functionals L 1 , . . . , L n and there is a δ > 0 such that whenever |L i (x)| < δ (i = 1, . . . , n), then x < 1.
Since H is of infinite dimension, there is an x = 0 such that L i (x) = 0 for every i = 1, . . . , n. (Otherwise every linear functionals would be a linear combination of L 1 , . . . , L n , and then H = H * would be finite dimensional.) Then λx ∈ U for every λ ∈ C and λx < 1 for every λ ∈ C, which is impossible.
In the example above the function f is a generalized polynomial with respect the discrete topology, and the only reason why it is not a generalized polynomial is the lack of continuity. We show that this is the case whenever the σ-ideals N σ GP (G) and N σ GP (H) are proper. Theorem 11. Let G, H be topological Abelian groups, and suppose that N σ GP (G) is a proper σ-ideal in G, and N σ GP (H) is a proper σ-ideal in H. If f : (G × H) → C is a separately generalized polynomial function, then f is a generalized polynomial with respect to the discrete topology.
Proof. Suppose f satisfies the conditions. By Lemma 7, it is enough to show that the degrees deg f x and f y are bounded.
Since N σ GP is a proper σ-ideal, there is an n such that A n / ∈ N GP . We fix such an n, and prove that ∆ (0,h 1 ) . . . ∆ (0,hn) f = 0 for every h 1 , . . . , h n ∈ H.
Let g denote the left hand side of (4). Then g(x, y) = s i=1 a i f (x, y + b i ), where s = 2 n , a i = ±1 and b i ∈ H for every i. Let y ∈ H be fixed. Then g y = s i=1 a i f y+b i , and thus g y is a generalized polynomial on G. If x ∈ A n , then deg f x < n, and thus g x = 0. Therefore g y (x) = 0 for every x ∈ A n . Since g y is a generalized polynomial and A n / ∈ N GP , it follows that g y = 0. Since y was arbitrary, this proves (4). Thus deg f x < n for every x ∈ G.
A similar argument shows that, for a suitable m, deg f y < m for every y ∈ H. Assuming somewhat stronger than being a Baire space we can omit the condition of continuity from Corollary 12.
Theorem 14. Let G, H be connected topological Abelian groups, and suppose that either (i) G and H are complete metric spaces, or (ii) G and H are locally compact.
If f : (G × H) → C is a separately generalized polynomial function, then f is a generalized polynomial.
Proof. Under the conditions the groups G, H are connected Baire spaces. By Theorem 11, f is a generalized polynomial with respect to the discrete topology. So we only have to prove that f is continuous.
Suppose (i). Then the function f is Baire 1 on G × H by [7, p. 378]. Since G × H is completely metrizable, it follows that f has a point of (joint) continuity. Now [12,Theorem 3.6] states that if f is a discrete generalized polynomial on an Abelian group which is generated by every neighbourhood of the origin, and if f has a point of continuity, then f is continuous everywhere. In our case the group G×H is connected, so the condition is satisfied, and we conclude that f is continuous everywhere on G × H.
If (ii) holds, then G, H are connected and locally compact Abelian groups, hence they are σ-compact as well. By [10,Theorem 1.2], it follows that f has a point of continuity, and then we can complete the proof as above.
Remark 15. The proof of Theorem 14 actually gives the following, more general statement.
Suppose that (i) G and H are connected, (ii) G / ∈ N σ GP (G) and H / ∈ N σ GP (H), and (iii) every separately continuous function on G×H has at least one point of joint continuity. Then every separately generalized polynomial function on G × H is a generalized polynomial.