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numtheory

 kronecker
 Inhomogeneous Diophantine approximation

 Calling Sequence kronecker(ineqs, xvars, yvars) kronecker(form, alpha, err)

Parameters

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

Description

 • Important: The numtheory[kronecker] command has been deprecated.  Use the superseding command NumberTheory[InhomogeneousDiophantine] 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

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

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

$|{a}_{\mathrm{m1}}{x}_{1}+\dots +{a}_{\mathrm{mn}}{x}_{n}-{\mathrm{\alpha }}_{m}-{y}_{m}|\le {\mathrm{err}}_{m}$

 or

$\mathrm{valuep}\left({a}_{11}{x}_{1}+\dots +{a}_{1n}{x}_{n}-{\mathrm{\alpha }}_{1}-{y}_{1},{p}_{1}\right)\le {\mathrm{err}}_{1}$

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

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

 • 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.
 • In the second calling sequence, if the $\mathrm{\alpha }$'s are all the same, the list $\left[{\mathrm{\alpha }}_{1},\mathrm{...},{\mathrm{\alpha }}_{m}\right]$ may be replaced by $\mathrm{\alpha }$. The err's may be similarly replaced in the real case.
 • The command with(numtheory,kronecker) allows the use of the abbreviated form of this command.

Examples

 > $\mathrm{with}\left(\mathrm{numtheory}\right):$
 > $\mathrm{with}\left(\mathrm{padic}\right):$
 > $\mathrm{kronecker}\left(\left\{\left|0.01\mathrm{log}\left(2\right)x+24\mathrm{log}\left(5\right)y-8{3}^{\frac{1}{2}}z-{ⅇ}^{2.5}-u\right|\le {10}^{-7},\left|-3.7{ⅇ}^{2}x+y+{3}^{\frac{1}{3}}z-{5}^{\frac{1}{3}}-v\right|\le {10}^{-3}\right\},\left\{x,y,z\right\},\left\{u,v\right\}\right)$
 $\left[{x}{=}{8026}\right]{,}\left[{y}{=}{-}{3174}\right]{,}\left[{z}{=}{6916}\right]{,}\left[{v}{=}{-}{212628}\right]{,}\left[{u}{=}{-}{218388}\right]$ (1)
 > $x≔'x':$$y≔'y':$$u≔'u':$$v≔'v':$
 > $\mathrm{kronecker}\left(\left\{\mathrm{valuep}\left(\mathrm{log}\left(5\right)x+\mathrm{log}\left(7\right)y-\mathrm{log}\left(5\right)-u,3\right)\le {3}^{-20},\mathrm{valuep}\left(\frac{1x}{\mathrm{log}\left(7\right)}+\mathrm{log}\left(11\right)y-\mathrm{log}\left(7\right)-v,5\right)\le {5}^{-15},\mathrm{valuep}\left(\mathrm{log}\left(3\right)x+{ⅇ}^{7}y-\mathrm{log}\left(3\right)-w,7\right)\le {7}^{-12}\right\},\left\{x,y\right\},\left\{u,v,w\right\}\right)$
 $\left[{x}{=}{-}{15516275}\right]{,}\left[{y}{=}{6404775}\right]{,}\left[{w}{=}{-}{9747866955}\right]{,}\left[{u}{=}{-}{1192024656}\right]{,}\left[{v}{=}{-}{27148890349}\right]$ (2)
 > $\mathrm{kronecker}\left(\left[\left[\mathrm{log}\left(2\right),\mathrm{log}\left(5\right),{3}^{\frac{1}{2}}\right],\left[{ⅇ}^{2},\mathrm{π},{3}^{\frac{1}{3}}\right]\right],\left[ⅇ,{2}^{\frac{1}{2}}\right],\left[{10}^{-2},{10}^{-5}\right]\right)$
 $\left[{-}{2863}{,}{-}{10057}{,}{-}{1494}\right]{,}\left[{-}{20761}{,}{-}{54906}\right]$ (3)
 > $\mathrm{kronecker}\left(\left[\left[\left[\mathrm{log}\left(3\right),\mathrm{log}\left(7\right),\mathrm{log}\left(13\right)\right],\left[\mathrm{sin}\left(5\right),\frac{1}{\mathrm{log}\left(7\right)},{ⅇ}^{5}\right]\right],\left[2,5\right]\right],\left[\mathrm{log}\left(5\right),\mathrm{log}\left(11\right)\right],\left[10,15\right]\right)$
 $\left[{-}{2000}{,}{3125}{,}{2825}\right]{,}\left[{-}{800}{,}{-}{26295606385}\right]$ (4)