Number of integers represented by families of binary forms III: fewnomials

2025/09/10 by Étienne Fouvry, Fouvry, Etienne, Michel Waldschmidt +1
Mathematics · #Advanced Differential Equations and Dynamical Systems #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Primary 11E76 #Secondary 11D45 11D85

paper · pdf · doi:10.48550/arxiv.2509.08335

openalex publication_date 2025/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In a series of papers we investigated the following question: given a family \calF of binary forms having nonzero discriminant and integer coefficients, for each d\geqslant 3, we estimate the number of integers m with |m|\leqslant N which are represented by an element in \calF of degree \geqslant d. Under suitable assumptions, asymptotically as N→∞, the main term in the estimate is given by the forms in \calF having degree d (if any), while the forms of degree >d contribute only to the error term. The present text is devoted to fewnomials a₀X^kr+a₁X^k(r-1)Y^k+⋯ +a_r-1X^kY^k(r-1)+a_rY^kr with fixed r\geqslant 1 and varying k,a₀,a₁,…,a_r.

Citations

• Number of integers represented by families of binary forms II: binomial forms

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