We report on a set of studies concerning the description of New Physics (NP) effects characterized by a scale much higher than the electroweak scale. We concentrate on the residual effects described by an effective lagrangian involving only bosonic fields and restrict to the case that the Higgs field is described linearly, so that the Higgs particle exists. It has already been emphasized that this lagrangian may be represented by eight $dim=6$ operators; namely the blind CP-conserving operators $\O_W$, $\O_{B\Phi}$, $\O_{W\Phi}$, $\O_{UB}$, $\O_{UW}$, the ''superblind'' ${\cal O}_{\Phi 2}$ and the two CP-violating ones $\ol{\O}_{UB}$ and $\ol{\O}_{UW}$. To each operator and for any given value of its coupling, we associate an unambiguous NP scale determined by the energy where unitarity is saturated. Our study of the possible NP tests realizable at a future linear $e^+e^-$ collider concentrates on $(e^+e^-\to HZ ~,~ H\gamma)$, and on $\gamma\gamma$ collisions producing boson pairs ($\gamma\gamma \to W^+W^-$, $ZZ$, $Z\gamma$, $\gamma\gamma$, $HH$), or a single Higgs ($\gamma\gamma \to H$). The sensitivities to the various operators involved in the effective lagrangian are estimated. We find that the testable NP scales vary from a few tens of TeV in the boson pair processes, up to 65 TeV in single H production. Ways to disentangle the various operators are also proposed. In this respect a comparison of the NP effects in the various boson pair processes, like \eg\@ a comparison of $e^+e^-\to HZ$ versus $e^+e^-\to H\gamma$, should be very fruitful. This is applied also to the study of the Higgs branching ratios, and especially the ratio $\Gamma(H\to\gamma\gamma)/\Gamma (H\to\gamma Z)$.

Comment: 13 pages with 15 figures included. (e-mail renard@lpmsun2.lpm.univ-montp2.fr)