In the watermark detection scenario, also known as zero-bit watermarking, a watermark, carrying no hidden message, is inserted in content. The watermark detector checks for the presence of this particular weak signal in content. The article looks at this problem from a classical detection theory point of view, but with side information enabled at the embedding side: the watermark signal is a function of the host content. Our study is twofold. The first issue is to design the best embedding function for a given detection function (a Neyman-Pearson detector structure is assumed). The second issue is to find the best detection function for a given embedding function. This yields two conditions, which are mixed into one 'fundamental' partial differential equation. Solutions of this fundamental equation are heavily dependent on the probability distribution function of the host signals. This conference paper is an extract of~\cite{Furon2006:A-constructive}, where we only look at white gaussian hosts. This gives birth to polynomials solutions known as Hermite polynomial, whose extension is the JANIS watermarking scheme, invented heuristically some years ago.