CHU’S 1303 IDENTITY IMPLIES BOMBIERI’S 1990 NORM-INEQUALITY [Via An Identity of Beauzamy and Dégot]

(Appeared in the Amer. Math. Monthly Nov. 1994.)

Doron ZEILBERGER^{1}

Blessed 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 deep inequality, A, follows from a mere equality B, which, in turn, follows from a more general, and trivial^{2} identity C.

In this note we demonstrate this, following [3], with A:= Bombieri’s norm inequality[2]^{3}, B:= an identity of Reznick[5], and C := an identity of Beauzamy and Dégot[3]. This exposition differs from the original only in the punch line: I give a 1-line proof of C, using Chu’s identity.

Let P (x_{1}, …, x_{n}) and Q(x_{1}, …, x_{n}) be two polynomials in n variables:

P = ∑_{i_{1},...,i_{n}≥0} a_{i_{1},...,i_{n}}x_{1}^{i_{1}} · . . . · x_{n}^{i_{n}} , Q = ∑_{i_{1},...,i_{n}≥0} b_{i_{1},...,i_{n}}x_{1}^{i_{1}} · . . . · x_{n}^{i_{n}} .

The Bombieri inner product[2] is defined by

[P, Q] := ∑_{i_{1},...,i_{n}≥0}(i_{1}! … i_{n}!) · a_{i_{1},...,i_{n}}b_{i_{1},...,i_{n}} ,

and the Bombieri norm, by: ∥P ∥ := \sqrt{[P, P]} .

Bombieri’s Inequality A: Let P and Q be any homogeneous polynomials in (x_{1}, …, x_{n}), then

∥P Q∥ ≥ ∥P ∥∥Q∥ .

In order to state B and C, we need to introduce the following notation. D_{i} := \frac{∂}{∂x_{i}}, (i = 1, …, n), P^{(i_{1},...,i_{n})} := D_{1}^{i_{1}} … D_{n}^{i_{n}}P, and for any polynomial A(x_{1}, …, x_{n}), A(D_{1}, …, D_{n}) denotes the linear partial differential operator with constant coefficients obtained by replacing x_{i} by D_{i}.

A follows almost immediately from([5][3]):

Reznick’s Identity B: For any polynomials P, Q in n variables:

∥P Q∥^{2} = ∑_{i_{1},...,i_{n}≥0} \frac{∥P^{(i_{1},...,i_{n})}(D_{1}, …, D_{n})Q(x_{1}, …, x_{n})∥^{2}}{i_{1}! · . . . · i_{n}!} .

Beauzamy and Dégot’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 for Advanced 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 to

have far reaching applications to computer algebra![1].

[P Q, RS] = ∑_{i_{1},...,i_{n}≥0} \frac{[R^{(i_{1},...,i_{n})}(D_{1}, …, D_{n})Q(x_{1}, …, x_{n}), P^{(i_{1},...,i_{n})}(D_{1}, …, D_{n})S(x_{1}, …, x_{n})]}{(i_{1}! … i_{n}!)} .

Proof of B ⇒ A: Pick the terms for which i_{1} + … + i_{n} equals the (total) degree of P, let’s call it p, and note that for those (i_{1}, …, i_{n}), P^{(i_{1},...,i_{n})}(x_{1}, …, x_{n}) = (i_{1}! … i_{n}!)a_{i_{1},...,i_{n}}, so

∑_{i_{1}+...+i_{n}=p} \frac{∥P^{(i_{1},...,i_{n})}(D_{1}, …, D_{n})Q(x_{1}, …, x_{n})∥^{2}}{i_{1}! · . . . · i_{n}!} = ∑_{i_{1}+...+i_{n}=p} ∥a_{i_{1},...,i_{n}}Q(x_{1}, …, x_{n})∥^{2} · (i_{1}! · . . . · i_{n}!)

= ( ∑_{i_{1}+...+i_{n}=p} (a_{i_{1},...,i_{n}})^{2} · (i_{1}! · . . . · i_{n}!)) ∥Q(x_{1}, …, x_{n})∥^{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 to be typical monomials, (P = x_{1}^{p_{1}} · . . . · x_{n}^{p_{n}}, and similarly for Q,R, and S), for which the assertion follows immediately by applying Chu’s[4] identity^{4}

∑_{i≥0} \binom{r}{i}\binom{s}{p − i} = \binom{r + s}{p} ,

to r = r_{t}, s = s_{t}, p = p_{t}, (t = 1 … n), (using i_{t} for i), and taking their product. Q.E.D.

References

1. B. Beauzamy, Products of polynomials and a priori estimates for coefficients in polynomial decompositions: 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 many variables, J. Number Theory 36 (1990), 219-245.

3. B. Beauzamy and J. Dégot, 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.

Added Sept. 24, 2017: Kai-Liang Lin just informed that the so-called Bombieri’s inequality should be called Neuberger’s inequality, as it was discovered, in 1974, by John M. Neuberger. John M. Neuberger, ”Norm of symmetric product compared with norm of tensor product”, Linear and Multilinear Algebra 2(1974), 115-121.

4 Rediscovered in the 18th century by Vandermonde. Proved by counting, in two different ways, the number of ways

of picking p lucky winners out of a set of r boys and s girls.