p-value interpretation in corrcoef for non-linear correlation

Dear Alfonso,

I'am not sure that I should write this comment on the thread you handled this week : http://www.mathworks.com/matlabcentral/answers/141942-corrcoef-p-value-interpretation

So, may I be so bold as to show you this strange result I've got today with corrcoeff by asking you directly a question.

I actually want to test/understand thoroughly this function before using it on real data. Here is my snippet :

N = 1000;X = rand(N,1) - 1/2;
Y = [X(:,1) X(:,1).*X(:,1)];
[r p] = corrcoef(Y)

Most of the time, p(1,2) value is quite larger than .05, which means that X is not linearly correlated with X^2. Rigorously, it's true : X is correlated with X^2, but not linearly.

So my question : should I understand that the statistical test result represented by the p matrix is not relevant to detect non linear correlation ?

All the best

sylvain