{"paper":{"title":"Vinogradov systems with a slice off","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Julia Brandes, Trevor D. Wooley","submitted_at":"2017-07-19T12:37:13Z","abstract_excerpt":"Let $I_{s,k,r}(X)$ denote the number of integral solutions of the modified Vinogradov system of equations $$x_1^j+\\ldots +x_s^j=y_1^j+\\ldots +y_s^j\\quad (\\text{$1\\le j\\le k$, $j\\ne r$}),$$ with $1\\le x_i,y_i\\le X$ $(1\\le i\\le s)$. By exploiting sharp estimates for an auxiliary mean value, we obtain bounds for $I_{s,k,r}(X)$ for $1\\le r\\le k-1$. In particular, when $s,k\\in \\mathbb N$ satisfy $k\\ge 3$ and $1\\le s\\le (k^2-1)/2$, we establish the essentially diagonal behaviour $I_{s,k,1}(X)\\ll X^{s+\\epsilon}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.06047","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":""},"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"}