  Predictions of the top-pair production cross section in electron-proton (ep)
collisions at the future circular collider hadron-electron (FCC-he) and the
large hadron electron collider (LHeC) in the leading order (LO) and the
next-to-leading order (NLO) approximations in perturbative quantum
chromodynamics (QCD) are presented. Numerical results for the top structure
functions and the ratio of structure functions are computed at the
renormalization scale $\mu^{2}=Q^{2}+4m^{2}_{t}$ in the collinear generalized
double asymptotic scaling (DAS) approach. These results are presented in terms
of the effective parameters of the parameterization of $F_{2}(x,Q^{2})$, where
relying on a Froissart-bounded. These quantitative results can be important for
future collider experiments at the center-of-mass energy frontier and the
improvement of the phenomenological models for the development of the cosmic
ray cascades in ultra-high-energy domain. We obtained the uncertainties because
of the power-law behavior $x^{-\Delta}$ of the collinear parton distribution
functions (PDFs), the factorization, and renormalization scales, on the top
properties at high inelasticity for the FCC-he center-of-mass (CM) energy at
the NLO approximation.
