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Suppose that P ( z, f ) is a difference polynomial of the form (1.1) and contains exactly one term of maximal total degree in f ( z ) and its shifts.
Moreover, we assume that U ( z, f ) contains just one term of maximal total degree in f ( z ) and its shifts.
If (H z,f)) contains just one term of maximal total degree, then for any small enough (varepsilon>0), m bigl r,P z,f) bigr)=O bigl(r^{rho-1+varepsilon} bigr)+S r,f).
If H ( z, f ) just contains one term of maximal total degree, then for any ε > 0, m ( r, P ( z, f ) ) = O ( r σ − 1 + ε ) + S ( r, f ).
If H zz, f ) contains just one term of maximal total degree, then for any ε > 0, m ( r, P ( z, f ) ) = O ( r ρ - 1 + ε ) + S ( r, f ), possibly outside of an exceptional set of finite logarithmic measure.
Moreover, we assume that all coefficients a λ ( z ) in (2.2) are small in the sense that T ( r, a λ ) = S ( r, f ) and that U ( z, f ) contains just one term of maximal total degree in f ( z ) and its shifts.
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