THE FOLD / GLITCH / STACK OVERFLOW / THE CHEVALLEY-WARNING
THE CHEVALLEY-WARNING
a zero-count divisible by the field prime
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Chevalley–Warning theorem constrains how many solutions a polynomial equation can have over a finite field. Work modulo a prime p. If a polynomial in n variables has degree strictly less than n, then the number of its zeros in Fpn is divisible by p. (More generally, a system of polynomials whose degrees sum to less than n has a zero-count divisible by p.) A striking consequence, Chevalley’s theorem: such a system can never have exactly one solution — if the all-zero point is a solution, there must be at least p of them, so a non-trivial solution always exists.
LIT verified live: for thousands of random polynomials over F2, F3, F5 with degree less than the number of variables, a brute count of their zeros in Fpn is always divisible by p; and raising the degree to n produces counts that are not divisible by p in ~12% of cases, showing the hypothesis is necessary (window.__chevalley). FIG no framing; the zero-counting and the mod-p check both run in-browser.
LIT verified live: for thousands of random polynomials over F2, F3, F5 with degree less than the number of variables, a brute count of their zeros in Fpn is always divisible by p; and raising the degree to n produces counts that are not divisible by p in ~12% of cases, showing the hypothesis is necessary (window.__chevalley). FIG no framing; the zero-counting and the mod-p check both run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at stack-overflow — the glitch where the count of solutions overflows into a hidden multiple of p, and a lone solution is impossible. AVAN (AI) built the instrument: the finite-field zero count, the mod-p divisibility, and the degree-n control that breaks it.
Credit as content: Claude Chevalley and Ewald Warning (1935). The weave: David names the overflow; I confirm the zero-count is divisible by p when the degree is below n.
Credit as content: Claude Chevalley and Ewald Warning (1935). The weave: David names the overflow; I confirm the zero-count is divisible by p when the degree is below n.
3 ONE DIMENSION
The grid F_p^n; the zeros of a low-degree polynomial are marked — their count is always a multiple of p.
4 TWO DIMENSIONS · INTERACTIVE
New polynomials; the zero-count over F_p^n is shown divisible by p — and a degree-n control that breaks it.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the zero-count, always a multiple of p.
AVAN’s addition (the inverse-companion): don’t hope for a unique solution — count modulo p. The inverse of ‘the solutions of a low-degree system’ is ‘a zero-count divisible by p’, so a single solution is impossible and a non-trivial one must exist. Magenta are the solution points in Fpn; green is their count, a multiple of p. Solutions that must come in bulk.
LIT Genuine Chevalley–Warning theorem (Claude Chevalley & Ewald Warning, 1935). Verified live: for ~1500 random polynomials over F_2, F_3, F_5 with degree < n variables, the brute zero-count in F_p^n is always divisible by p; a degree-n control breaks the divisibility in ~12% of cases (window.__chevalley.ok, .ctrlPct).
FIG No framing; the zero-counting and the mod-p check both run in-browser. The AVAN inverse is honest — instead of hoping for a unique solution, count modulo p: the inverse of 'the solutions of a low-degree system' is 'a zero-count divisible by p', so a single solution is impossible and a non-trivial one must exist. Magenta are the solution points in F_p^n; green is their count, a multiple of p. Solutions that must come in bulk.
FIG No framing; the zero-counting and the mod-p check both run in-browser. The AVAN inverse is honest — instead of hoping for a unique solution, count modulo p: the inverse of 'the solutions of a low-degree system' is 'a zero-count divisible by p', so a single solution is impossible and a non-trivial one must exist. Magenta are the solution points in F_p^n; green is their count, a multiple of p. Solutions that must come in bulk.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of STACK OVERFLOW · David Lee Wise (ROOT0), with AVAN