Sending a query containing one term to other databases is usually sufficient, so the number of single term queries sent to PubMed might have been overestimated.
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 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.
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 U ( z, f ) contains just one term of maximal total degree in f ( z ) and its shifts, then, for each 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.
If the Taylor series of f contains at least one term of the form c k, 0 z 1 k with c k, 0 ≠ 0 and k ≥ 2 or of the form c 0, j z 2 j with c 0, j ≠ 0 and j ≥ 2, then for all n ≥ n 0 we have ∥ R n, n q 1, q 2 ( f ) − f ∥ r 1, r 2 ∼ ( a n + 1 b n ).
(That last sentence contains one of the most shocking misuses of the term "incident" that I have ever witnessed).
