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CHU'S 1303 IDENTITY IMPLIES BOMBIERI'S 1990 NORM-INEQUALITY
 
[Via An Identity of Beauzamy and D\'egot]
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\rm
(Appeared in the {\it Amer. Math. Monthly}  Nov. 1994.)
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\it
\qquad\qquad
\qquad\qquad\qquad\qquad\qquad\qquad Doron ZEILBERGER\footnote{$^1$}
{\eightrm 
Department of Mathematics, Temple University,
Philadelphia, PA 19122, USA. 
{\eighttt zeilberg@math.temple.edu .} Supported 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.
}
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\qquad\qquad\qquad\qquad\qquad{\it Blessed are the meek: for they shall
inherit the earth (Matthew V.5)}
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\noindent
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\noindent
\rm
Inequalities are deep, while {\it equalities} are shallow. 
Nevertheless, it sometimes happens that a deep
inequality, {\bf A}, follows from a mere {\it equality} {\bf B}, which, in 
turn, follows from a more general, and {\it trivial}\footnote{$^2$}
{\eightrm Trivial to verify, not to conceive!} identity {\bf C}. 
\medskip
In this note we demonstrate this, following [3],
with {\bf A}$:=$ Bombieri's 
norm inequality[2]\footnote{$^3$}
{\eightrm 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].
},
{\bf B}$:=$ an identity of Reznick[5], and {\bf C} $:=$ an identity of Beauzamy 
and D\'egot[3]. This exposition differs from the original only in the
punch line: I give a 1-line proof of {\bf C}, using Chu's identity.
 
\medskip
Let $P(x_1 , \dots , x_n)$ and $Q(x_1 , \dots , x_n)$ be two 
polynomials in $n$ variables:
 
$$
P= \sum_{ i_1 , \dots , i_n  \geq 0 } a_{ i_1 , \dots , i_n}
x_1^{i_1} \cdot \dots \cdot x_n^{i_n} \quad , \quad
Q= \sum_{i_1 , \dots , i_n  \geq 0 } b_{ i_1 , \dots , i_n}
x_1^{i_1} \cdot \dots \cdot x_n^{i_n}  \quad .
$$
 
The {\it Bombieri inner product}[2] is defined by
 
$$
[P,Q]:=
 \sum_{ i_1 , \dots , i_n  \geq 0 }( {i_1}! \dots {i_n}! ) \cdot
a_{ i_1 , \dots , i_n}
b_{ i_1 , \dots , i_n} \quad,
$$
 
and the {\it Bombieri norm}, by: $\Vert P \Vert := \sqrt{[P,P]} \quad .$
\medskip
{\bf Bombieri's Inequality A}: Let $P$ and $Q$ be any {\it homogeneous} 
polynomials in $(x_1 , \dots , x_n)$, then
 
$$
\Vert PQ \Vert \geq  \Vert P \Vert \Vert Q \Vert \quad .
$$
\medskip
In order to state {\bf B} and {\bf C}, we need to introduce the 
following notation.
$D_i := {{\partial} \over {\partial x_i}}$, ($i= 1, \dots ,n$),
$P^{( i_1 , \dots , i_n )} := D_1^{i_1} \dots D_n^{i_n} P$, and
for any polynomial $A(x_1, \dots , x_n)$, $A(D_1 , \dots , D_n)$
denotes the linear partial differential operator with constant
coefficients obtained by replacing $x_i$ by $D_i$.
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{\bf A} follows almost immediately from([5][3]):
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{\bf Reznick's Identity B:} For any polynomials $P$, $Q$ in $n$ variables:
 
$$
\Vert P Q \Vert ^2 =
 \sum_{i_1 , \dots , i_n \geq 0 } 
{{\Vert P^{(i_1 , \dots , i_n )} (D_1 , \dots , D_n) Q(x_1 , \dots , x_n ) 
\Vert^2} \over {i_1! \cdot \dots \cdot i_n!}} \quad.
$$
 
{\bf Beauzamy and D\'egot's Identity C:}
For any polynomials $P$,$Q$,$R$,$S$ in $n$ variables:
 
$$
[PQ,RS]=
\sum_{i_1 , \dots  , i_n  \geq 0} 
{{[R^{(i_1 , \dots , i_n)}(D_1 , \dots , D_n) Q(x_1 , \dots , x_n) ,
P^{(i_1 , \dots , i_n)}(D_1 , \dots , D_n) S(x_1 , \dots , x_n) ]}
\over
{( i_1! \dots i_n! )}} \quad.
$$
 
{\bf Proof of  B} $\Rightarrow$ {\bf A}:
Pick the terms for which  $ i_1 + \dots + i_n$ equals the (total)
degree of $P$, let's call it $p$, and note
that for those  $( i_1 , \dots , i_n )$,
$P^{( i_1 , \dots , i_n )} (x_1 , \dots , x_n ) =( i_1! \dots i_n!)
a_{i_1 , \dots , i_n}$, so
 
$$
\sum_{i_1 + \dots + i_n = p} 
{{\Vert P^{(i_1 , \dots , i_n )} (D_1 , \dots , D_n) Q(x_1 , \dots , x_n ) 
\Vert^2} \over {i_1! \cdot \dots \cdot i_n!}} 
=\sum_{i_1 + \dots + i_n = p} 
{{\Vert a_{i_1 , \dots , i_n} Q(x_1 , \dots , x_n ) 
\Vert^2} \cdot {( i_1! \cdot \dots \cdot i_n! ) }} 
$$
$$
=
\left ( \sum_{i_1 + \dots + i_n = p} 
{{ ( a_{i_1 , \dots , i_n})^2 } \cdot {( i_1! \cdot \dots \cdot i_n!) }} 
\right )
\Vert Q(x_1 , \dots , x_n ) 
\Vert^2 =  \Vert P \Vert^2 \Vert Q \Vert^2 \quad.
$$
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{\bf Proof of C} $\Rightarrow$ {\bf B:} Take $R=P$
and $S=Q$.
\medskip
{\bf 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,
($P=x_1^{p_1} \cdot \dots \cdot x_n^{p_n}$, and similarly for
$Q$,$R$, and $S$), for which
the assertion follows immediately  by applying Chu's[4]
identity\footnote{$^4$}
{\eightrm 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.}
 
$$
\sum_{i \geq 0} { {r} \choose {i}} {{s} \choose {p-i}} = {{r+s} \choose {p}}
\quad ,
$$
 
to $r=r_t , s=s_t, p=p_t, (t= 1 \dots n)$, (using $i_t$ for $i$),
and taking their product. Q.E.D.
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\noindent
{\bf References}
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\noindent
1. B. Beauzamy, {\it Products of polynomials and a priori estimates
for coefficients in polynomial decompositions: A sharp result},
J. Symbolic Computation {\bf 13}(1992), 463-472.
 
2. B. Beauzamy, E. Bombieri, P. Enflo, and H.L. Montgomery,
{\it Products of polynomials in many variables}, J. Number Theory {\bf 36}
(1990), 219-245.
 
3. B. Beauzamy and  J. D\'egot, {\it 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, {\it An inequality for products of polynomials}, Proc.
Amer. Math. Soc. {\bf 117}(1993), 1063-1073.
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Nov. 3, 1993 ; Revised: April 5, 1994.

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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 {\bf 2}(1974), 115-121.

\bye
 
 
