A Stothers–Mason theorem with a difference radical

Differential calculus is not a unique way to observe polynomial equations such as $$a+b=c$$ a + b = c . We propose a way of applying difference calculus to estimate multiplicities of the roots of the polynomials a, b and c satisfying the equation above. Then a difference abc theorem for polynomials is proved using a new notion of a radical of a polynomial. Results, for example, on the non-existence of polynomial solutions to difference Fermat and difference Super-Fermat functional equations are given as applications. We also introduce a truncated second main theorem for differences, and use it to consider these functional equations with non-polynomial entire solutions. Equations with polynomial or non-polynomial solutions are observed to see the sharpness of results obtained.

Degeneracy second main theorem for meromorphic mappings and moving hypersurfaces with truncated counting functions and applications

In this paper, we establish a new second main theorem for meromorphic mappings of [Formula: see text] into [Formula: see text] and moving hypersurfaces with truncated counting functions in the case, where the meromorphic mappings may be algebraically degenerate. A version of the second main theorem with weighted counting functions is also given. Our results improve the recent results on this topic. As an application, an algebraic dependence theorem for meromorphic mappings sharing moving hypersurfaces is given.

A SECOND MAIN THEOREM FOR ENTIRE CURVES IN A PROJECTIVE VARIETY WHOSE DERIVATIVES VANISH ON INVERSE IMAGE OF HYPERSURFACE TARGETS

We establish a second main theorem for algebraically nondegenerate entire curves f in a projective variety V ⊂ P n(C) and a hypersurface target {D1, D2, . . . , Dq} satisfying f∗,z = 0 for all z ∈ ∪q j=1f−1(Dj).

Degenerated second main theorem for holomorphic curves into algebraic varieties

In this paper, under the refinement of the subgeneral position, we give an improvement for the Second Main Theorem with truncated counting functions of algebraically non-degenerate holomorphic curves into algebraic varieties [Formula: see text] intersecting divisors in subgeneral position with some index.