  We forecast dark energy constraints that could be obtained from a new large
sample of Type Ia supernovae where those at high redshift are acquired with the
Euclid space mission. We simulate a three-prong SN survey: a z<0.35 nearby
sample (8000 SNe), a 0.2<z<0.95 intermediate sample (8800 SNe), and a
0.75<z<1.55 high-z sample (1700 SNe). The nearby and intermediate surveys are
assumed to be conducted from the ground, while the high-z is a joint ground-
and space-based survey. This latter survey, the "Dark Energy Supernova
Infra-Red Experiment" (DESIRE), is designed to fit within 6 months of Euclid
observing time, with a dedicated observing program. We simulate the SN events
as they would be observed in rolling-search mode by the various instruments,
and derive the quality of expected cosmological constraints. We account for
known systematic uncertainties, in particular calibration uncertainties
including their contribution through the training of the supernova model used
to fit the supernovae light curves. Using conservative assumptions and a 1-D
geometric Planck prior, we find that the ensemble of surveys would yield
competitive constraints: a constant equation of state parameter can be
constrained to sigma(w)=0.022, and a Dark Energy Task Force figure of merit of
203 is found for a two-parameter equation of state. Our simulations thus
indicate that Euclid can bring a significant contribution to a purely
geometrical cosmology constraint by extending a high-quality SN Hubble diagram
to z~1.5. We also present other science topics enabled by the DESIRE Euclid
observations
