{"ID":23595659,"CreatedAt":"2026-09-18T05:42:42.291333095Z","UpdatedAt":"2026-09-18T05:42:42.291333095Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.20354","arxiv_id":"2609.20354","title":"A complete classification of permutation binomials of the form $X^r(X^{q-1}+a)$ over finite fields","abstract":"We classify, for every prime power $q$ and every $e\\geqslant2$, the permutation binomials $X^r(X^{q-1}+a)$ over $\\mathbb F_{q^e}$. Writing $\\ell_j(q)=(q^j-1)/(q-1)$, such a binomial is a permutation if and only if $\\gcd(r,q-1)=1$, $(-a)^{\\ell_e(q)}\\ne1$, and $r\\ell_h(q)\\equiv1\\pmod{\\ell_e(q)}$ for some $1\\leqslant h\u003ce$ coprime to $e$. This proves a conjecture of Masuda, Rubio, and Santiago: every permutation binomial of this form arises from $(X^{q^h}+aX)\\circ X^r$ for a suitable $h$. We also determine the exact number of distinct permutation functions represented by this family. As a further consequence, we completely classify the broader family $X^r(X^{d(q-1)}+a)$ in the coprime-index case $\\gcd(d,\\ell_e(q))=1$. The new ingredient in the main classification is the necessity argument: selected Hermite power sums are organized so that Lucas' theorem turns their coefficients into digit conditions; a Farey-guided local argument then forces successive base-$q$ digits, and cyclic rotations yield the inverse congruence. In characteristic $2$, a mod-$4$ lift to an auxiliary ring retains endpoint information lost modulo $2$.","short_abstract":"We classify, for every prime power $q$ and every $e\\geqslant2$, the permutation binomials $X^r(X^{q-1}+a)$ over $\\mathbb F_{q^e}$. Writing $\\ell_j(q)=(q^j-1)/(q-1)$, such a binomial is a permutation if and only if $\\gcd(r,q-1)=1$, $(-a)^{\\ell_e(q)}\\ne1$, and $r\\ell_h(q)\\equiv1\\pmod{\\ell_e(q)}$ for some $1\\leqslant h\u003ce$...","url_abs":"https://arxiv.org/abs/2609.20354","url_pdf":"https://arxiv.org/pdf/2609.20354v1","authors":"[\"Xiang Fan\"]","published":"2026-09-17T13:18:18Z","proceeding":"math.CO","tasks":"[\"math.CO\",\"math.NT\"]","methods":"[\"Generative Adversarial Network\"]","has_code":false}