Expand description
The real closure of the values of several infinitesimals: √(dt + dx), ∛(h + dt), every root the
program’s arithmetic reaches among the values of more than one infinitesimal, held exactly.
A value of dx, declared much smaller than dt, that is no rational function of a root of dx
over the values of the larger infinitesimals is an AlgValue: the recipe that built it, a
squarefree polynomial over those rational functions with the value among its roots, and the value’s
expansion in dx with exact coefficients (crate::expansion). The polynomial composes as the
value does — α + β is a root of Res_z(P(z), Q(y − z)), α·β of Res_z(P(z), z^n·Q(y/z)), 1/α
of P reversed, ⁿ√α of P(y^n), α^k of Res_z(P(z), y − z^k) — and decides what the expansion
alone cannot. The orders of P’s nonzero roots are the slopes of its Newton polygon, so a value
whose expansion vanishes through the largest of them is the root 0, and a nonzero value’s leading
term lies at or below it: every comparison is a bounded scan of exact coefficients.
One form for a value. A value that is a rational function of a root of dx is held as that
function (Multi::Over): its candidate is the Padé approximant of its expansion within the
degrees Gauss’s lemma bounds a root of the monic polynomial by, kept only when it is a root of the
polynomial and its difference from the value is the root 0. A value prints as that function, as a
radical when a power of it is one (√(dt + dx), 2·√(dt + dx), 1/√(dt + dx)), or by its two
leading terms and the order of the rest (1/(2·√dt) - (1/(8·dt^(3/2)))·dx + O(dx^2)).
Structs§
- AlgValue
- A value of the real closure of several infinitesimals at its smallest. See the module documentation.