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

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

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.

$\mathrm{with}\left(\mathrm{numtheory}\right)\:$

$\mathrm{with}\left(\mathrm{padic}\right)\:$

$\mathrm{minkowski}\left(\left\{\mathrm{abs}\left(3^{\frac{1}{3}}\mathrm{z1}+\mathrm{\pi }\mathrm{z2}-\mathrm{s2}\right)\le {10}^{-4}\,\mathrm{abs}\left(\exp \left(1\right)\mathrm{z1}+2^{\frac{1}{2}}\mathrm{z2}-\mathrm{s1}\right)\le {10}^{-2}\right\}\,\left\{\mathrm{z1}\,\mathrm{z2}\right\}\,\left\{\mathrm{s1}\,\mathrm{s2}\right\}\right)$

$\left[\mathrm{z1}=7484\right],\left[\mathrm{z2}=\mathrm{-2534}\right],\left[\mathrm{s2}=2833\right],\left[\mathrm{s1}=16760\right]$

$\mathrm{minkowski}\left(\left\{\mathrm{abs}\left(\exp \left(2\right)x+\mathrm{\pi }y+3^{\frac{1}{3}}z-s\right)\le {10}^{-5}\,\mathrm{abs}\left(\log \left(2\right)x+\log \left(5\right)y+3^{\frac{1}{2}}z-r\right)\le {10}^{-2}\,\mathrm{valuep}\left(\log \left(3\right)x+\log \left(7\right)y+\log \left(13\right)z-u\,2\right)\le 2^{-7}\,\mathrm{valuep}\left(\sin \left(5\right)x+\frac{1}{\log \left(7\right)}y+\exp \left(5\right)z-v\,5\right)\le 5^{-9}\right\}\,\left\{x\,y\,z\right\}\,\left\{r\,s\,u\,v\right\}\right)$

$\left[x=\mathrm{-154525}\right],\left[y=\mathrm{-165325}\right],\left[z=147450\right],\left[s=\mathrm{-1448518}\right],\left[r=\mathrm{-117798}\right],\left[u=\mathrm{-52}\right],\left[v=14613\right]$

$\mathrm{minkowski}\left(\left[\left[\exp \left(1\right)\right]\right]\,{10}^{-2}\right)$

$\left[465\right],\left[1264\right]$

$\mathrm{minkowski}\left(\left[\left[\exp \left(1\right)\right]\right]\,{10}^{-2}\right)$

$\left[465\right],\left[1264\right]$

$\mathrm{minkowski}\left(\left[\left[\left[\log \left(5\right)\,\log \left(7\right)\right]\,\left[\frac{1}{\log \left(7\right)}\,\log \left(11\right)\right]\,\left[\log \left(3\right)\,\exp \left(7\right)\right]\right]\,\left[3\,5\,7\right]\right]\,\left[20\,15\,12\right]\right)$

$\left[328700\,11704900\right],\left[395598\,70177\,\mathrm{-334904}\right]$