aa r X i v : . [ m a t h . GN ] J a n Separately polynomial functions
Gergely Kiss and Mikl´os Laczkovich (Budapest, Hungary)January 11, 2021
Abstract
It is known that if f : R → R is a polynomial in each variable,then f is a polynomial. We present generalizations of this fact, when R is replaced by G × H , where G and H are topological Abeliangroups. We show, e.g., that the conclusion holds (with generalizedpolynomials in place of polynomials) if G is a connected Baire spaceand H has a dense subgroup of finite rank or, for continuous functions,if G and H are connected Baire spaces. The condition of continuitycan be omitted if G and H are locally compact or complete metricspaces. We present several examples showing that the results are notfar from being optimal. It was proved by F. W. Carroll in [2] that if f : R → R is a polynomialin each variable, then f is a polynomial. Our aim is to find generalizationsof this fact, when R is replaced by the product of two topological Abeliangroups. Keywords: polynomials, generalized polynomials, functions on product spaces MR subject classification: Both authors were supported by the Hungarian National Foundation for ScientificResearch, Grant No. K124749.
1n topological Abelian groups we distinguish between the class of poly-nomials and the wider class of generalized polynomials (see the next sectionfor the definitions). The two classes coincide if the group contains a densesubgroup of finite rank. Now, the scalar product on the square of a Hilbertspace is an example of a continuous function which is a polynomial in eachvariable without being a polynomial (see Example 1 below). Therefore, theappropriate problem is to find conditions on the groups G and H ensur-ing that whenever a function on G × H is a generalized polynomial in eachvariable, then it is a generalized polynomial.We show that this is the case if G is not the union of countable many zerosets 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 connectedBaire 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 functionon G × H is a generalized polynomial in each variable, then it is a generalizedpolynomial (Corollary 12).It is not clear if the condition of continuity can be omitted from Corollary12 (see Question 13). The problem is that a generalized polynomial must becontinuous by definition, and a separately continuous function on the productof Baire spaces can be discontinuous everywhere, as it was shown recentlyin [9]. In our case, however, there are some extra conditions: the spacesare also connected, and the function in question is a generalized polynomial.It is conceivable that continuity follows under these conditions. As for thebiadditive case, see [3].We show that if G and H are connected complete metric spaces, or theyare connected and locally compact, then every separately generalized poly-nomial function on G × H is a generalized polynomial (Theorem 14).The proof shows that the conclusion holds whenever G and H are con-nected Baire spaces and such that every separately continuous function on G × H has at least one point of joint continuity.There are several topological conditions implying this property. In fact,the topic has a vast literature starting with the paper [10]. See, e.g., thepapers [4], [5], [6], [11]. 2 Preliminaries
Let G be a topological Abelian group. We denote the group operation byaddition, and denote the unit by 0. The translation operator T h and thedifference operator ∆ h are defined by T h f ( x ) = f ( x + h ) and ∆ h f ( x ) = f ( x + h ) − f ( x ) for every f : G → C and h, x ∈ G .We say that a continuous function f : G → C is a generalized polynomial ,if there is an n ≥ h . . . ∆ h n +1 f = 0 for every h , . . . , h n +1 ∈ G .The smallest n with this property is the degree of f , denoted by deg f . Thedegree of the identically zero function is −
1. We denote by GP = GP G theset of generalized polynomials defined on G .A function f : G → C is said to be a polynomial , if there are continuousadditive functions a , . . . , a n : G → C and there is a P ∈ C [ x , . . . , x n ] suchthat f = P ( a , . . . , a n ). It is well-known that every polynomial is a general-ized polynomial. It is also easy to see that the linear span of the translatesof a polynomial is of finite dimension. More precisely, a function is a poly-nomial if and only if it is a generalized polynomial, and the linear span ofits 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 : X → C of f are defined by f x ( y ) = f y ( x ) = f ( x, y )( x ∈ X, y ∈ Y ).Let G, H be topological Abelian groups. A function f : ( G × H ) → C isa 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 separatelygeneralized 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 functionon 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 Hilbertspace. Then the scalar product f ( x, y ) = h x, y i on G is a separately poly-nomial 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 every3 , y ∈ G . Thus the sections of f are polynomials.Now, while the scalar product is a generalized polynomial (of degree 2)on G , it is not a polynomial on G , because the dimension of the linear spanof its translates is infinite. Indeed, let g ( x ) = h x, x i = k x k for every x ∈ G .Then ∆ h g ( x ) = 2 h h, x i + k h k for every h ∈ G . It is easy to see that thefunctions h h, x i ( h ∈ G ) generate a linear space of infinite dimension, andthen 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 ageneralized polynomial.We denote by r ( G ) the torsion free rank of the group G ; that is, thecardinality of a maximal independent system of elements of G of infiniteorder. Thus r ( G ) = 0 if and only if G is torsion. In the sequel by therank 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 ofgeneralized polynomials on G coincide (see [8, Theorem 9]).The set of roots of a function f : G → C is denoted by Z f . That is, Z f = { x ∈ G : f ( x ) = 0 } . We put N P = N P ( G ) = { A ⊂ G : ∃ p ∈ P G , p = 0 , A ⊂ Z p } and N GP = N GP ( G ) = { A ⊂ G : ∃ p ∈ GP G , p = 0 , A ⊂ Z p } . It is easy to see that N P and N GP are proper ideals of subsets of G . Let N σP and N σGP denote the σ -ideals generated by N P and N GP , respectively. Notethat N P ⊂ N GP and N σP ⊂ N σGP .If G is discrete, then N σP and N σGP are not proper σ -ideals (except when G is torsion), according to the next observation. Proposition 2.
Let G be a discrete Abelian group. If G is not torsion, then G ∈ N σP . 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 , the4dditive 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.Then p r = ψ + r is a nonzero polynomial on G for every r ∈ Q . If x ∈ G ,then x is the root of p r , where r = − ψ ( x ) ∈ Q . Therefore, G = S r ∈ Q Z p r ∈N σP . (cid:3) A simple sufficient condition for
G / ∈ N σGP is given by the next result.
Lemma 3. If G is a connected Baire space, then the σ -ideals N σP and N σGP are proper; that is, G / ∈ N σP and G / ∈ N σGP . Proof.
It is enough to prove that every element of N GP is nowhere dense.Suppose A ∈ N GP is dense in a nonempty open set U . Let p ∈ GP ( G ) bea nonzero generalized polynomial vanishing on A . Since A ⊂ Z p and Z p is closed, we have U ⊂ Z p . Since G is connected, every neighbourhood ofthe origin generates G . It is known that in such a group, if a generalizedpolynomial vanishes on a nonempty open set, then it vanishes everywhere(see [12, Theorem 3.2, p. 33]). This implies that p is identically zero, whichis impossible. (cid:3) Our next result generalizes Carroll’s theorem [2].
Theorem 4.
Let
G, H be topological Abelian groups, and suppose that (i) N σGP ( G ) is a proper σ -ideal in G , and (ii) 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 conditionthat G is a connected Baire space. 5 emma 6. Let H be a topological Abelian group, and suppose that H hasa dense subgroup of finite rank. Then, for every positive integer d , thereare finitely many points x , . . . , x s ∈ H and there are generalized polynomials q , . . . , q s ∈ GP H of degree < d such that p = P si =1 p ( x i ) · q i for every p ∈ GP H with deg p < d . Proof.
Let GP Let f : ( G × H ) → C be a separately generalizedpolynomial function. Put G n = { x ∈ G : deg f x < n } ( n = 1 , , . . . ). Since N σGP ( G ) is a proper σ -ideal in G , there is an n such that G n / ∈ N GP ( G ). Fixsuch an n .By Lemma 3, there are points y , . . . , y s ∈ H and generalized polynomials q , . . . , q s ∈ GP H such that p = P si =1 p ( y i ) · q i for every p ∈ GP H withdeg p < n . Therefore, we have f ( x, y ) = s X i =1 f ( x, y i ) q i ( y )7or every x ∈ G n and y ∈ H . If y ∈ H is fixed, then f ( x, y ) − P si =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 fol-lows that f ( x, y ) = P si =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 . (cid:3) 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 ofTheorem 4 may fail. We shall need the easy direction of the following result. Lemma 7. Let G, H be discrete Abelian groups. A function f : ( G × H ) → C is a generalized polynomial if and only if the sections f x ( x ∈ G ) and f y ( y ∈ H ) are generalized polynomials of bounded degree. Proof. Suppose f : ( G × H ) → C is a generalized polynomial of degree < d .Then ∆ ( x , . . . ∆ ( x d , f = 0 for every x , . . . , x d ∈ G . Then, for every y ∈ H ,we have ∆ ( x , . . . ∆ ( x d , f y = 0 for every x , . . . , x d ∈ G , and thus f y is ageneralized polynomial of degree < d for every y ∈ H . A similar argumentshows 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∆ ( h , . . . ∆ ( h d , f y = 0 (1)for every h , . . . , h d ∈ G and y ∈ H , and∆ (0 ,k ) . . . ∆ (0 ,k d ) f x = 0 (2)for every k , . . . , k d ∈ H and x ∈ G . In order to prove that f is a generalizedpolynomials of degree < d , it is enough to show that∆ ( a ,b ) . . . ∆ ( a d ,b d ) f = 0 (3)for every ( a i , b i ) ∈ G × H ( i = 1 , . . . , d ). The identity ∆ u + v = T u ∆ v + ∆ u gives ∆ ( a i ,b i ) = T ( a i , ∆ (0 ,b i ) + ∆ ( a i , for every i . Therefore, the left hand side of (3) is the sum of terms of theform T c ∆ c . . . ∆ c d f , where c ∈ G × { } , and c i ∈ ( G × { } ) ∪ ( { } × H ) for8very i . If there are at least d indices i with c i ∈ ( G × { } ), then (1) gives∆ c . . . ∆ c d f = 0. Otherwise there are at least d indices i with c i ∈ ( { }× H ),and then (2) gives ∆ c . . . ∆ c d f = 0. This proves (3). (cid:3) Now we turn to the first example. Example 8. Let G, H be discrete Abelian groups. We show that if none of G and H is torsion, then there is a separately polynomial function f : ( G × H ) → C such that f is not a generalized polynomial on G × H .By Proposition 2, N σP ( G ) is not a proper σ -ideal; that is, G = S ∞ 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 = 0and A n ⊂ Z p n . Then p n is not constant; that is, deg p n ≥ P n = p · · · p n ; then P n ( x ) = 0 for every x ∈ S ni =1 A i , and we havedeg P < deg P < . . . . (Here we use the fact that deg pq = deg p + deg q forevery p, q ∈ GP G , p, q = 0.) Note that for every x ∈ G we have P n ( x ) = 0for all but a finite number of indices n .Similarly, we find polynomials Q n ∈ P H such that deg Q < deg Q < . . . ,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 ) = P ∞ 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, wehave f x ∈ P H for every x ∈ G .The degrees deg f y ( y ∈ H ) are not bounded. Indeed, for every N , thereis an y ∈ H be such that Q N ( y ) = 0. Then f y = P Mn =1 Q n ( y ) · P n withan 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 positiveand finite rank, then the conclusion of Theorem 4 fails. That is, G / ∈ N σGP ( 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 showthat if G is a topological Abelian group such that P G contains nonconstant9olynomials, 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 infiniteorder, where κ ≥ ω . Let K denote the subgroup of H generated by theelements h α ( α < κ ). Every element of K is of the form P α<κ k α h α , where k α ∈ Z for every α , and all but a finite number of the coefficients k α equalzero.Let p ∈ P G be a nonconstant polynomial. We define f ( x, y ) = P ∞ i =1 k i · p i ( x ) for every x ∈ G and y ∈ K , y = P α<κ k α h α . (Note that the sum onlycontains a finite number of nonzero terms for every x and y .) In this way wedefined 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 wedefine f ( x, y ) = n · f ( x, ny ) for every x ∈ G . It is easy to see that f ( x, y ) iswell-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 n · P Ni =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 degreesdeg 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 , , . . . ).In the example above we may choose G in such a way that G / ∈ N σGP ( G )holds. (Take, e.g., G = R .) In our next example this condition holds forboth G and H . Example 10. Let E be a Banach space of infinite dimension, and let G bethe 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 σGP ( G ). Indeed, the original norm topology of E is stronger than τ , and makes E a connected Baire space. If a functionis 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 thus10 P ( G ) ⊂ N P ( E ) and N σP ( G ) ⊂ N σP ( E ). Since N σP ( E ) is proper by Lemma3, it follows that N σP ( G ) is proper. The same is true for N σGP .Now let H be an infinite dimensional Hilbert space, and let G be the ad-ditive group of H equipped with the weak topology of H . Let f be the scalarproduct on H . Since the linear functionals and conjugate linear functionalsare continuous w.r.t. the weak topology, it follows that f is a separatelypolynomial function on G (see Example 1).However, f is not a generalized polynomial on G , since f is not contin-uous. In order to prove this, it is enough to show that f ( x, x ) = k x k is notcontinuous on H w.r.t. the weak topology. Suppose it is. Then there is aneighbourhood U of 0 such that k x k < x ∈ U . By the definitionof the weak topology, there are linear functionals L , . . . , L n and there is a δ > | L i ( x ) | < δ ( i = 1 , . . . , n ), then k x k < H is of infinite dimension, there is an x = 0 such that L i ( x ) = 0for every i = 1 , . . . , n . (Otherwise every linear functionals would be a linearcombination of L , . . . , L n , and then H = H ∗ would be finite dimensional.)Then λx ∈ U for every λ ∈ C and k λx k < λ ∈ C , which isimpossible.In the example above the function f is a generalized polynomial withrespect the discrete topology, and the only reason why it is not a generalizedpolynomial is the lack of continuity. We show that this is the case wheneverthe σ -ideals N σGP ( G ) and N σGP ( H ) are proper. Theorem 11. Let G, H be topological Abelian groups, and suppose that N σGP ( G ) is a proper σ -ideal in G , and N σGP ( H ) is a proper σ -ideal in H .If f : ( G × H ) → C is a separately generalized polynomial function, then f isa generalized polynomial with respect to the discrete topology. Proof. Suppose f satisfies the conditions. By Lemma 7, it is enough toshow that the degrees deg f x and f y are bounded.Put A n = { x ∈ G : deg f x < n } . Then G = S ∞ n =1 A n . Since N σGP is aproper σ -ideal, there is an n such that A n / ∈ N GP . We fix such an n , andprove that ∆ (0 ,h ) . . . ∆ (0 ,h n ) f = 0 (4)11or every h , . . . , h n ∈ H .Let g denote the left hand side of (4). Then g ( x, y ) = P si =1 a i f ( x, y + b i ),where s = 2 n , a i = ± b i ∈ H for every i . Let y ∈ H be fixed. Then g y = P si =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 forevery x ∈ A n . Since g y is a generalized polynomial and A n / ∈ N GP , it followsthat 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 . (cid:3) Corollary 12. Let G, H be topological Abelian groups, and suppose that G and H are connected Baire spaces. If f : ( G × H ) → C is a continuous sep-arately generalized polynomial function, then f is a generalized polynomial. (cid:3) Question 13. Is the condition of continuity necessary in the statement ofCorollary 12? (See the introduction.)Assuming somewhat stronger than being a Baire space we can omit thecondition of continuity from Corollary 12. Theorem 14. Let G, H be connected topological Abelian groups, and supposethat 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 isa 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 discretetopology. So we only have to prove that f is continuous.12uppose (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 generalizedpolynomial on an Abelian group which is generated by every neighbourhoodof the origin, and if f has a point of continuity, then f is continuous every-where. 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. (cid:3) Remark 15. The proof of Theorem 14 actually gives the following, moregeneral statement.Suppose that (i) G and H are connected, (ii) G / ∈ N σGP ( G ) and H / ∈N σGP ( H ), and (iii) every separately continuous function on G × H has at leastone point of joint continuity. Then every separately generalized polynomialfunction on G × H is a generalized polynomial. References [1] J. Acz´el and , J. Dhombres: Functional equations in several variables. Encyclopedia of Mathematics and its Applications, 31. Cambridge Uni-versity Press, Cambridge, 1989.[2] F. W. Carroll, A polynomial in each variable separately is a polynomial, Amer. Math. Monthly (1961), 42.[3] J.P.R. Christensen and P. Fischer, Joint continuity of measurable biad-ditive mappings, Proc. Amer. Math. Soc. (1988), no. 4, 1125-1128.[4] K.C. Ciesielski and D. Miller, A continuous tale on continuous andseparately continuous functions, Real Anal. Exchange (2016), no. 1,19-54.[5] M. Henriksen and R.G. 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