CHU’S 1303 IDENTITY IMPLIES BOMBIERI’S 1990 NORM-INEQUALITY [Via
영국의 수학자 보미에리는 1990년에 n개의 변수를 가진 모든 유한 항 다항식 P와 Q에 대해 |PQ| ≤ √(∥P∥^2 × ∥Q∥^2)라는 제약 조건을 증명했다. 이 논문에서는 Chu의 정리를 사용하여 Bombeiery의 제약 조건을 부정하는 예를 찾았다.
Bombeiery의 제약 조건은 다음을 만족해야 한다: |PQ| ≥ √(∥P∥^2 × ∥Q∥^2)이다. 이 논문에서는 Beauzamy와 D´egot의 정리를 사용하여 Bombeiery의 제약 조건이 항상 참인지 부정했다.
Beauzamy와 D´egot의 정리는 두 변수를 가진 모든 다항식 P, Q, R, S에 대해 다음을 만족한다: [PQ, RS] = ∑[R(i)(D)Q(x), P(i)(D)S(x)]. 이 정리를 사용하여 Bombeiery의 제약 조건이 항상 참인지 부정했다.
Chu의 정리는 다음과 같다: ∑(i ≥ 0) r^i s^(p-i) = (r+s)^p. 이 정리를 사용하여 Beauzamy와 D´egot의 정리를 증명했다.
결과적으로, Bombeiery의 제약 조건이 항상 참인지 부정한 결과가 나왔다.
영어 요약 시작:
The paper is a discussion on Bombieri's norm inequality (1990) which states that for any homogeneous polynomials P and Q in n variables, |PQ| ≥ √(∥P∥^2 × ∥Q∥^2). The author attempts to provide a counterexample by using Chu's identity.
Chu's identity is used to prove Beauzamy and D´egot's identity which states that for any polynomials P, Q, R, S in n variables, [PQ, RS] = ∑[R(i)(D)Q(x), P(i)(D)S(x)]. This identity is then used to show that Bombieri's inequality does not always hold.
The paper concludes by stating that the author has been able to provide a counterexample to Bombieri's norm inequality.
CHU’S 1303 IDENTITY IMPLIES BOMBIERI’S 1990 NORM-INEQUALITY [Via
arXiv:math/9307202v1 [math.CO] 2 Jul 1993CHU’S 1303 IDENTITY IMPLIES BOMBIERI’S 1990 NORM-INEQUALITY [ViaAn Identity of Beauzamy and D´egot](Appeared in the Amer. Math.
Monthly 101(1994), 894-896..)Doron ZEILBERGER1Blessed are the meek: for they shall inherit the earth (Matthew V.5)Inequalities are deep, while equalities are shallow. Nevertheless, it sometimes happens that a deepinequality, A, follows from a mere equality B, which, in turn, follows from a more general, andtrivial2 identity C.In this note we demonstrate this, following [3], with A:= Bombieri’s norm inequality[2]3, B:= anidentity of Reznick[5], and C := an identity of Beauzamy and D´egot[3].
This exposition differsfrom the original only in the punch line: I give a 1-line proof of C, using Chu’s identity.Let P(x1, . .
. , xn) and Q(x1, .
. .
, xn) be two polynomials in n variables:P =Xi1,...,in≥0ai1,...,inxi11 · . .
. · xinn,Q =Xi1,...,in≥0bi1,...,inxi11 · .
. .
· xinn.The Bombieri inner product[2] is defined by[P, Q] :=Xi1,...,in≥0(i1! .
. .
in!) · ai1,...,inbi1,...,in,and the Bombieri norm, by: ∥P∥:=p[P, P].Bombieri’s Inequality A: Let P and Q be any homogeneous polynomials in (x1, .
. .
, xn), then∥PQ∥≥∥P∥∥Q∥.In order to state B and C, we need to introduce the following notation. Di :=∂∂xi , (i = 1, .
. .
, n),P (i1,...,in) := Di11 . .
. Dinn P, and for any polynomial A(x1, .
. .
, xn), A(D1, . .
. , Dn) denotes the linearpartial differential operator with constant coefficients obtained by replacing xi by Di.A follows almost immediately from([5][3]):Reznick’s Identity B: For any polynomials P, Q in n variables:∥PQ∥2 =Xi1,...,in≥0∥P (i1,...,in)(D1, .
. .
, Dn)Q(x1, . .
. , xn)∥2i1!
· . .
. · in!.Beauzamy and D´egot’s Identity C: For any polynomials P,Q,R,S in n variables:1 Department of Mathematics, Temple University, Philadelphia, PA 19122, USA.
zeilberg@math.temple.edu . Sup-ported in part by the NSF.
This note was written while the author was on leave (Fall 1993) at the Institute forAdvanced Study, Princeton. I would like to thank Don Knuth for a helpful suggestion.2 Trivial to verify, not to conceive!3 It was needed by Beauzamy and Enflo in their research on deep questions on Banach spaces.
It also turned out tohave far reaching applications to computer algebra! [1].1
[PQ, RS] =Xi1,...,in≥0[R(i1,...,in)(D1, . .
. , Dn)Q(x1, .
. .
, xn), P (i1,...,in)(D1, . .
. , Dn)S(x1, .
. .
, xn)](i1! .
. .
in! ).Proof of B ⇒A: Pick the terms for which i1 + .
. .
+ in equals the (total) degree of P, let’s call itp, and note that for those (i1, . .
. , in), P (i1,...,in)(x1, .
. .
, xn) = (i1! .
. .
in! )ai1,...,in, soXi1+...+in=p∥P (i1,...,in)(D1, .
. .
, Dn)Q(x1, . .
. , xn)∥2i1!
· . .
. · in!=Xi1+...+in=p∥ai1,...,inQ(x1, .
. .
, xn)∥2 · (i1! · .
. .
· in! )=Xi1+...+in=p(ai1,...,in)2 · (i1!
· . .
. · in!
)∥Q(x1, . .
. , xn)∥2 = ∥P∥2∥Q∥2.Proof of C ⇒B: Take R = P and S = Q.Proof of C: Both sides are linear in P, in Q, in R, and in S, so it suffices to take them all tobe typical monomials, (P = xp11 · .
. .
· xpnn , and similarly for Q,R, and S), for which the assertionfollows immediately by applying Chu’s[4] identity4Xi≥0ri sp −i=r + sp,to r = rt, s = st, p = pt, (t = 1 . .
. n), (using it for i), and taking their product.
Q.E.D.References1.B. Beauzamy, Products of polynomials and a priori estimates for coefficients in polynomialdecompositions: A sharp result, J.
Symbolic Computation 13(1992), 463-472.2. B. Beauzamy, E. Bombieri, P. Enflo, and H.L.
Montgomery, Products of polynomials in manyvariables, J. Number Theory 36 (1990), 219-245.3.
B. Beauzamy and J. D´egot, Differential Identities, I.C.M., Paris, preprint.4. Chu Chi-kie, manuscript, 1303, China.
(See J. Needham, Science and Civilization in China, v.3, Cambridge University Press, New York, 1959.)5. B. Reznick, An inequality for products of polynomials, Proc.
Amer. Math.
Soc. 117(1993),1063-1073.Nov.
3, 1993 ; Revised: April 5, 1994.4 Rediscovered in the 18th century by Vandermonde. Proved by counting, in two different ways, the number of waysof picking p lucky winners out of a set of r boys and s girls.2
출처: arXiv:9307.202 • 원문 보기