numtheory(deprecated)/minkowski - Maple Help

numtheory(deprecated)

 minkowski
 solve Minkowski's linear forms (homogeneous diophantine approximation)

 Calling Sequence minkowski(ineqs, xvars, yvars) minkowski(form, err)

Parameters

 ineqs - inequality or a set of inequalities with abs and/or valuep (p-adic valuation) xvars - variable or a set of variables yvars - variable or a set of variables form - list of lists of real numbers and/or p-adic numbers and primes err - real number or a list of real numbers or list of positive integers

Description

 • Important: The numtheory package has been deprecated.  Use the superseding command NumberTheory[HomogeneousDiophantine] instead.
 • This function finds a solution ${x}_{1},{x}_{2},\dots ,{x}_{n},{y}_{1},\dots ,{y}_{m}$ over the integers to a set of inequalities of the form

$|{a}_{11}{x}_{1}+\dots +{a}_{1n}{x}_{n}-{y}_{1}|\le {\mathrm{err}}_{1}$

$\mathrm{..............}$

$|{a}_{\mathrm{j1}}{x}_{1}+\dots +{a}_{\mathrm{jn}}{x}_{n}-{y}_{j}|\le {\mathrm{err}}_{j}$

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

$\mathrm{..............}$

$\mathrm{valuep}\left({a}_{m1}{x}_{1}+\dots +{a}_{\mathrm{mn}}{x}_{n}-{y}_{m},{p}_{m}\right)\le {\mathrm{err}}_{m}$

 where $0\le j\le m$.
 • The inequalities can be described either explicitly, corresponding to the first calling sequence shown above (see the first two examples below) or implicitly, corresponding to the second calling sequence (see the last two examples below).
 • If the first calling sequence is used (i.e., the inequalities are given explicitly), then the result is returned in the form

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

 If the second calling sequence is used, the result is returned as a pair of lists, the first corresponding to the x values and the second corresponding to the y values.
 • The command with(numtheory,minkowski) allows the use of the abbreviated form of this command.

Examples

Important: The numtheory package has been deprecated.  Use the superseding command NumberTheory[HomogeneousDiophantine] instead.

 > $\mathrm{with}\left(\mathrm{numtheory}\right):$
 > $\mathrm{with}\left(\mathrm{padic}\right):$
 > $\mathrm{minkowski}\left(\left\{\mathrm{abs}\left({3}^{\frac{1}{3}}\mathrm{z1}+\mathrm{\pi }\mathrm{z2}-\mathrm{s2}\right)\le {10}^{-4},\mathrm{abs}\left(\mathrm{exp}\left(1\right)\mathrm{z1}+{2}^{\frac{1}{2}}\mathrm{z2}-\mathrm{s1}\right)\le {10}^{-2}\right\},\left\{\mathrm{z1},\mathrm{z2}\right\},\left\{\mathrm{s1},\mathrm{s2}\right\}\right)$
 $\left[{\mathrm{z1}}{=}{7484}\right]{,}\left[{\mathrm{z2}}{=}{-2534}\right]{,}\left[{\mathrm{s2}}{=}{2833}\right]{,}\left[{\mathrm{s1}}{=}{16760}\right]$ (1)
 > $\mathrm{minkowski}\left(\left\{\mathrm{abs}\left(\mathrm{exp}\left(2\right)x+\mathrm{\pi }y+{3}^{\frac{1}{3}}z-s\right)\le {10}^{-5},\mathrm{abs}\left(\mathrm{log}\left(2\right)x+\mathrm{log}\left(5\right)y+{3}^{\frac{1}{2}}z-r\right)\le {10}^{-2},\mathrm{valuep}\left(\mathrm{log}\left(3\right)x+\mathrm{log}\left(7\right)y+\mathrm{log}\left(13\right)z-u,2\right)\le {2}^{-7},\mathrm{valuep}\left(\mathrm{sin}\left(5\right)x+\frac{1}{\mathrm{log}\left(7\right)}y+\mathrm{exp}\left(5\right)z-v,5\right)\le {5}^{-9}\right\},\left\{x,y,z\right\},\left\{r,s,u,v\right\}\right)$
 $\left[{x}{=}{-154525}\right]{,}\left[{y}{=}{-165325}\right]{,}\left[{z}{=}{147450}\right]{,}\left[{s}{=}{-1448518}\right]{,}\left[{r}{=}{-117798}\right]{,}\left[{u}{=}{-52}\right]{,}\left[{v}{=}{14613}\right]$ (2)
 > $\mathrm{minkowski}\left(\left[\left[\mathrm{exp}\left(1\right)\right]\right],{10}^{-2}\right)$
 $\left[{465}\right]{,}\left[{1264}\right]$ (3)
 > $\mathrm{minkowski}\left(\left[\left[\mathrm{exp}\left(1\right)\right]\right],{10}^{-2}\right)$
 $\left[{465}\right]{,}\left[{1264}\right]$ (4)
 > $\mathrm{minkowski}\left(\left[\left[\left[\mathrm{log}\left(5\right),\mathrm{log}\left(7\right)\right],\left[\frac{1}{\mathrm{log}\left(7\right)},\mathrm{log}\left(11\right)\right],\left[\mathrm{log}\left(3\right),\mathrm{exp}\left(7\right)\right]\right],\left[3,5,7\right]\right],\left[20,15,12\right]\right)$
 $\left[{328700}{,}{11704900}\right]{,}\left[{395598}{,}{70177}{,}{-334904}\right]$ (5)