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Recently, an analogue overF  q    [  T  ]of Landau's theorem on sums of two squares was considered by Bary‐Soroker, Smilansky and Wolf. They counted the number of monic polynomials inF  q    [  T  ]of degree   n   of the formA  2   +  T   B  2    , which we denote by    B  (  n  ,  q  )   . They studied    B  (  n  ,  q  )    in two limits: fixed   n   and large   q  ; and fixed   q   and large   n  . We generalize their result to the most general limitq  n   →  ∞   . More precisely, we proveB  (  n  ,  q  )  ∼   K   q    ⋅    (n  −   1  2    n    )    ⋅   q   n    ,q   n    →  ∞  ,for an explicit constantK  q   =  1  +  O   (  1  /  q  )    . Our methods are different and are based on giving explicit bounds on the coefficients of generating functions. These methods also apply to other problems, related to polynomials with prime factors of even degree.

A POLYNOMIAL ANALOGUE OF LANDAU'S THEOREM AND RELATED PROBLEMS