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NumberTheory

 HomogeneousDiophantine
 solution to Minkowski's linear forms

Parameters

 ineqs - inequality or set of inequalities with abs or valuep xvars - name or set of names yvars - name or set of names real_cfs, padic_cfs - convertible to a Matrix of real numbers adicities - convertible to a Vector of prime numbers real_errors - convertible to a Vector of real numbers padic_errors - convertible to a Vector of positive integers

Description

 • The HomogeneousDiophantine function finds a solution ${x}_{1},\dots ,{x}_{n},{y}_{1},\dots ,{y}_{m}$ over the integers to a set of inequalities of the form

$\left|{a}_{1,1}{x}_{1}+{a}_{1,n}{x}_{n}+\mathrm{...}-{y}_{1}\right|\le {\mathrm{err}}_{1}$

$\mathrm{...}$

$\left|{a}_{j,1}{x}_{1}+{a}_{j,n}{x}_{n}+\mathrm{...}-{y}_{j}\right|\le {\mathrm{err}}_{j}$

 or

$\mathrm{padic}:-\mathrm{valuep}\left({a}_{j+1,1}{x}_{1}+{a}_{j+1,n}{x}_{n}+\mathrm{...}-{y}_{j+1},{p}_{j+1}\right)\le {p}_{j+1}^{-{\mathrm{err}}_{j+1}}$

$\mathrm{...}$

$\mathrm{padic}:-\mathrm{valuep}\left({a}_{m,1}{x}_{1}+{a}_{m,n}{x}_{n}+\mathrm{...}-{y}_{m},{p}_{m}\right)\le {p}_{m}^{-{\mathrm{err}}_{m}}$

 where $\mathrm{padic}:-\mathrm{valuep}$ is the p-adic valuation.
 • The inequalities can be described explicitly, corresponding to the first calling sequence, or implicitly, corresponding to the other calling sequences.
 • If the first calling sequence is used, then the return value is of the form

$\left[{x}_{1}={s}_{1},\mathrm{...},{x}_{n}={s}_{n},{y}_{1}={t}_{1},\mathrm{...},{y}_{m}={t}_{m}\right]$

 • If the other calling sequences are used, then the return value is a two-element list corresponding to the x values and the y values,

$\left[\left[{s}_{1},\mathrm{...},{s}_{n}\right],\left[{t}_{1},\mathrm{...},{t}_{m}\right]\right]$

Examples

 > $\mathrm{with}\left(\mathrm{NumberTheory}\right):$
 > $\mathrm{HomogeneousDiophantine}\left(\left\{\left|\sqrt{2}x-y\right|\le {10}^{-3}\right\},\left\{x\right\},\left\{y\right\}\right)$
 $\left[{x}{=}{5741}{,}{y}{=}{8119}\right]$ (1)
 > $\mathrm{with}\left(\mathrm{padic}\right):$
 > $\mathrm{HomogeneousDiophantine}\left(\left\{\left|ⅇ\mathrm{z1}+{2}^{\frac{1}{2}}\mathrm{z2}-\mathrm{s1}\right|\le {10}^{-2},\left|{3}^{\frac{1}{3}}\mathrm{z1}+\mathrm{Pi}\mathrm{z2}-\mathrm{s2}\right|\le {10}^{-4}\right\},\left\{\mathrm{z1},\mathrm{z2}\right\},\left\{\mathrm{s1},\mathrm{s2}\right\}\right)$
 $\left[{\mathrm{z1}}{=}{1014}{,}{\mathrm{z2}}{=}{-5300}{,}{\mathrm{s2}}{=}{-15188}{,}{\mathrm{s1}}{=}{-4739}\right]$ (2)

An equivalent matrix form calling sequence is:

 > $\mathrm{HomogeneousDiophantine}\left(\left[\left[ⅇ,{2}^{\frac{1}{2}}\right],\left[{3}^{\frac{1}{3}},\mathrm{Pi}\right]\right],\left[{10}^{-2},{10}^{-4}\right]\right)$
 $\left[\left[{7484}{,}{-2534}\right]{,}\left[{16760}{,}{2833}\right]\right]$ (3)

The solutions may be different but both are valid.

Both abs and valuep may be used in the same system.

 > $\mathrm{HomogeneousDiophantine}\left(\left\{\left|\mathrm{log}\left(2\right)x+\mathrm{log}\left(5\right)y+{3}^{\frac{1}{2}}z-r\right|\le {10}^{-2},\mathrm{valuep}\left(\mathrm{sin}\left(5\right)x+\frac{1y}{\mathrm{log}\left(7\right)}+{ⅇ}^{5}z-v,5\right)\le {5}^{-9}\right\},\left\{x,y,z\right\},\left\{r,v\right\}\right)$
 $\left[{x}{=}{-2000}{,}{y}{=}{-8475}{,}{z}{=}{1600}{,}{r}{=}{-12255}{,}{v}{=}{214}\right]$ (4)

The error list for the p-adic cases are negatives of the exponents on the adicities.

 > $\mathrm{HomogeneousDiophantine}\left(\left[\left[\mathrm{log}\left(2\right),\mathrm{log}\left(5\right),{3}^{\frac{1}{2}}\right]\right],\left[{10}^{-2}\right],\left[\left[\mathrm{sin}\left(5\right),\frac{1}{\mathrm{log}\left(7\right)},{ⅇ}^{5}\right]\right],\left[5\right],\left[9\right]\right)$
 $\left[\left[{-3050}{,}{-2175}{,}{4450}\right]{,}\left[{2093}{,}{-13}\right]\right]$ (5)

Compatibility

 • The NumberTheory[HomogeneousDiophantine] command was introduced in Maple 2016.