Ideals and differential operators in the ring of polynomials of infinitely many variables
- PDF / 182,207 Bytes
- 11 Pages / 439.37 x 666.142 pts Page_size
- 50 Downloads / 213 Views
Ideals and differential operators in the ring of polynomials of infinitely many variables M. Laczkovich
Published online: 26 September 2014 © Akadémiai Kiadó, Budapest, Hungary 2014
Abstract Let be an uncountable and algebraically closed field. We prove that every ideal of the polynomial ring R = [x1 , x2 , . . .] is the intersection of ideals of the form { f ∈ R : D( f g)(c) = 0 for every g ∈ R}, where D is a differential operator of locally finite order, and c is a vector with values in . Keywords Ideals of rings of polynomials · Polynomials of infinitely many variables · Krull’s theorem · Differential operators Mathematics Subject Classification
13F20 · 13F25
1 Introduction and main results Let J be an ideal of the polynomial ring C[x1 , . . . , xn ]. It is known that for every f ∈ C[x 1 , . . . , x n ] \ J there exist a differential operator D and a complex vector c = (c1 , . . . , cn ) such that Dp(c) = 0 for every p ∈ J and D f (c) = 0. This statement is attributed to Krull by Lefranc in [5, p. 1953]. Lefranc used this result to prove that spectral synthesis holds on Zn ; that is, if V is a translation invariant linear space of functions mapping Zn into C, and if V is closed under pointwise convergence, then V is spanned by functions of the form p(x1 , . . . , xn ) · ec1 x1 +···+cn xn , where p ∈ C[x1 , . . . , xn ]. Krull’s theorem does not hold in the polynomial ring C[x 1 , x2 , . . .] as long as we mean by a differential operator a finite sum of the form
a(n 1 , . . . , n k ) ·
∂ n 1 +···+n k n . ∂ xin11 . . . ∂ xikk
M. Laczkovich (B) Department of Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/C, Budapest 1117, Hungary e-mail: [email protected] M. Laczkovich Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK
123
110
M. Laczkovich
Consider the following example. Let J denote the ideal of C[x 1 , x2 , . . .] generated by the polynomials x12 and x1 − xk (k = 2, 3, . . .). It is easy to check that x1 ∈ / J . Suppose there is a differential operator D and there is a vector c = (c1 , c2 , . . .) such that D( p)(c) = 0 for every p ∈ J , and D(x1 )(c) = 0. One can show that the vector c must be a root of J (see Proposition 3.2 in Sect.3). Since the only root of J is the zero vector, we must have N c = 0. Let D = a0 D0 + i=1 ai ∂∂xi + E, where D0 is the identity operator, and E is a differential operator in which every term is of degree at least two. If k ≤ N then we have 0 = D(x1 − xk )(0) = a1 − ak , and thus a1 = · · · = a N . If k > N then we have 0 = D(x1 − xk )(0) = a1 , and thus ai = 0 for every i ≤ N . Then D(x1 )(0) = 0, which is a contradiction. However, if we allow differential defined by suitable infinite series, the situation ∞ operators ∂ becomes different. Let D = i=1 . Then Dp makes sense for every p ∈ C[x1 , x2 , . . .], ∂ xi
since ∂∂xi p = 0 for all but a finite number of the indices i. It is easy to check that in the example above we have Dp(0) = 0 for every p ∈ J , but D(x1 )(0) = 0. Our aim is to show that this example i
Data Loading...
