Let
P be an
m -homogeneous polynomial in
n -complex variables
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si1.gif&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=d14882934c16a217cb876931f2ad3c9d" title ="Click to view the MathML source">x1 ,…,xn class="mathContainer hidden">class="mathCode">x 1 , … , x n . Clearly,
P has a unique representation in
the form
class="formula" id="fm0010">
and
the m -form
class="formula" id="fm0020">
satisfies
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si4.gif&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=c495f160e154179b08c98b0b41709c92" title ="Click to view the MathML source">LP (x,…,x)=P(x) class="mathContainer hidden">class="mathCode">L P ( x , … , x ) = P ( x ) for every
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si21.gif&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=51e24c7c4f844e450d0fc0c78a333a7a" title ="Click to view the MathML source">x∈Cn class="mathContainer hidden">class="mathCode">x ∈ C n . We show that, although
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si6.gif&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=e4cb0fb37682b5107f09ff783edacefc" title ="Click to view the MathML source">LP class="mathContainer hidden">class="mathCode">L P in general is non-symmetric, for a large
class of reasonable norms
class="mathmlsrc">title="View the MathML source" class ="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si212.gif&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=e23746b6c197461583f071dafac35731"> class="imgLazyJSB inlineImage" height="17" width="26" alt="View the MathML source" title ="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16302414-si212.gif"> the MathML source" title ="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16302414-si212.gif"> class="mathContainer hidden">class="mathCode">‖ ⋅ ‖ on
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si19.gif&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=847d41fe858350f8184d8533ad8f9dd1" title ="Click to view the MathML source">Cn class="mathContainer hidden">class="mathCode">C n the norm of
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si6.gif&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=e4cb0fb37682b5107f09ff783edacefc" title ="Click to view the MathML source">LP class="mathContainer hidden">class="mathCode">L P on
class="mathmlsrc">title="View the MathML source" class ="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si9.gif&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=e26457f8dec697251cee531644bf3870"> class="imgLazyJSB inlineImage" height="17" width="79" alt="View the MathML source" title ="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16302414-si9.gif"> the MathML source" title ="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16302414-si9.gif"> class="mathContainer hidden">class="mathCode">( C n , ‖ ⋅ ‖ ) m up to a logarithmic term
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si10.gif&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=22c4d36893e9c3576ac3897fd6e65ce7" title ="Click to view the MathML source">(clogn)m2 class="mathContainer hidden">class="mathCode">( c log n ) m 2 can be estimated by
the norm of
P on
class="mathmlsrc">title="View the MathML source" class ="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si11.gif&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=994c7cebb58fe9ae6d866fabaf63e5b2"> class="imgLazyJSB inlineImage" height="17" width="66" alt="View the MathML source" title ="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16302414-si11.gif"> the MathML source" title ="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16302414-si11.gif"> class="mathContainer hidden">class="mathCode">( C n , ‖ ⋅ ‖ ) ; here
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si12.gif&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=930e6572fc7b46b1f8431e45a204f058" title ="Click to view the MathML source">c≥1 class="mathContainer hidden">class="mathCode">c ≥ 1 denotes a universal constant. Moreover, for
the class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si13.gif&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=dd37c333e43863f7b1bfd6a9e351210a" title ="Click to view the MathML source">ℓp class="mathContainer hidden">class="mathCode">ℓ p -norms
class="mathmlsrc">title="View the MathML source" class ="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si14.gif&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=4bda895b2e18297f29ce4d865eeedf42"> class="imgLazyJSB inlineImage" height="17" width="34" alt="View the MathML source" title ="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16302414-si14.gif"> the MathML source" title ="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X16302414-si14.gif"> class="mathContainer hidden">class="mathCode">‖ ⋅ ‖ p ,
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16302414&_mathId=si15.gif&_user=111111111&_pii=S0022247X16302414&_rdoc=1&_issn=0022247X&md5=13ce8527e3073507043352050fe22260" title ="Click to view the MathML source">1≤p<2 class="mathContainer hidden">class="mathCode">1 ≤ p < 2 the logarithmic term in
the number
n of variables is even superfluous.