Let
P be an
m -homo
g eneous polynomial in
n -complex variables
g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si1.g if&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=d14882934c16a217cb876931f2ad3c9d" title="Click to view the MathML source">x1 ,…,xn g="si1.g if" overflow="scroll">x 1 , … , x n . Clearly,
P has a unique representation in the form
and the
m -form
satisfies
g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si4.g if&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=c495f160e154179b08c98b0b41709c92" title="Click to view the MathML source">LP (x,…,x)=P(x) g="si4.g if" overflow="scroll">L P ( x , … , x ) = P ( x ) for every
g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si21.g if&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=51e24c7c4f844e450d0fc0c78a333a7a" title="Click to view the MathML source">x∈Cn g="si21.g if" overflow="scroll">x ∈ C n . We show that, althou
g h
g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si6.g if&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=e4cb0fb37682b5107f09ff783edacefc" title="Click to view the MathML source">LP g="si6.g if" overflow="scroll">L P in
g eneral is non-symmetric, for a lar
g e class of reasonable norms
g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si212.g if&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=e23746b6c197461583f071dafac35731">g class="img LazyJSB inlineImag e" heig ht="17" width="26" alt="View the MathML source" title="View the MathML source" src="/sd/g rey_pxl.g if" data-inlimg eid="1-s2.0-S0022247X16302414-si212.g if">g heig ht="17" border="0" style="vertical-alig n:bottom" width="26" alt="View the MathML source" title="View the MathML source" src="http://orig in-ars.els-cdn.com/content/imag e/1-s2.0-S0022247X16302414-si212.g if"> g="si212.g if" overflow="scroll">‖ ⋅ ‖ on
g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si19.g if&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=847d41fe858350f8184d8533ad8f9dd1" title="Click to view the MathML source">Cn g="si19.g if" overflow="scroll">C n the norm of
g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si6.g if&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=e4cb0fb37682b5107f09ff783edacefc" title="Click to view the MathML source">LP g="si6.g if" overflow="scroll">L P on
g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si9.g if&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=e26457f8dec697251cee531644bf3870">g class="img LazyJSB inlineImag e" heig ht="17" width="79" alt="View the MathML source" title="View the MathML source" src="/sd/g rey_pxl.g if" data-inlimg eid="1-s2.0-S0022247X16302414-si9.g if">g heig ht="17" border="0" style="vertical-alig n:bottom" width="79" alt="View the MathML source" title="View the MathML source" src="http://orig in-ars.els-cdn.com/content/imag e/1-s2.0-S0022247X16302414-si9.g if"> g="si9.g if" overflow="scroll">( C n , ‖ ⋅ ‖ ) m up to a lo
g arithmic term
g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si10.g if&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=22c4d36893e9c3576ac3897fd6e65ce7" title="Click to view the MathML source">(clog n)m2 g="si10.g if" overflow="scroll">( c log n ) m 2 can be estimated by the norm of
P on
g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si11.g if&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=994c7cebb58fe9ae6d866fabaf63e5b2">g class="img LazyJSB inlineImag e" heig ht="17" width="66" alt="View the MathML source" title="View the MathML source" src="/sd/g rey_pxl.g if" data-inlimg eid="1-s2.0-S0022247X16302414-si11.g if">g heig ht="17" border="0" style="vertical-alig n:bottom" width="66" alt="View the MathML source" title="View the MathML source" src="http://orig in-ars.els-cdn.com/content/imag e/1-s2.0-S0022247X16302414-si11.g if"> g="si11.g if" overflow="scroll">( C n , ‖ ⋅ ‖ ) ; here
g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si12.g if&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=930e6572fc7b46b1f8431e45a204f058" title="Click to view the MathML source">c&g e;1 g="si12.g if" overflow="scroll">c &g e; 1 denotes a universal constant. Moreover, for the
g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si13.g if&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=dd37c333e43863f7b1bfd6a9e351210a" title="Click to view the MathML source">ℓp g="si13.g if" overflow="scroll">ℓ p -norms
g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si14.g if&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=4bda895b2e18297f29ce4d865eeedf42">g class="img LazyJSB inlineImag e" heig ht="17" width="34" alt="View the MathML source" title="View the MathML source" src="/sd/g rey_pxl.g if" data-inlimg eid="1-s2.0-S0022247X16302414-si14.g if">g heig ht="17" border="0" style="vertical-alig n:bottom" width="34" alt="View the MathML source" title="View the MathML source" src="http://orig in-ars.els-cdn.com/content/imag e/1-s2.0-S0022247X16302414-si14.g if"> g="si14.g if" overflow="scroll">‖ ⋅ ‖ p ,
g" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si15.g if&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=13ce8527e3073507043352050fe22260" title="Click to view the MathML source">1≤p<2 g="si15.g if" overflow="scroll">1 ≤ p < 2 the lo
g arithmic term in the number
n of variables is even superfluous.