{"paper":{"title":"On the number of solutions of systems of diagonal equations through diagonal GP-graphs: the general and the Hermitian-form cases","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Denis E. Videla, Ricardo A. Podest\\'a","submitted_at":"2026-08-19T23:49:29Z","abstract_excerpt":"For any $m, s \\in \\mathbb{N}$, we study the number $N_{m\\times s,q}(\\kappa, \\beta)$ of solutions $(x_1,\\ldots,x_s) \\in (\\mathbb{F}_q)^s$ of the monic system of diagonal equations\n  $$ X_{1}^{k_i} + \\cdots + X_{s}^{k_i}= \\beta_i, \\qquad (1\\le i \\le m), $$ with $\\kappa=(k_1,\\ldots,k_m) \\in \\mathbb{N}^m$ and $\\beta=(\\beta_1,\\ldots,\\beta_m) \\in (\\mathbb{F}_q)^m$. We show that this number can be obtained in terms of some data of \\textit{diagonal} GP-graphs $\\Gamma(\\kappa,q)$. This is a new family of graphs that we introduce here, i.e. Cayley graphs of the form\n  $$ \\Gamma(\\kappa,q) = Cay(\\mathbb{F}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.19507","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2608.19507/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}