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

or   

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

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,

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

$\mathrm{HomogeneousDiophantine}\left(\left\{\mathrm{abs}\left(\mathrm{sqrt}\left(2\right)x-y\right)\le {10}^{-3}\right\}\,\left\{x\right\}\,\left\{y\right\}\right)$

$\left[x=5741\,y=8119\right]$

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

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

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

$\mathrm{HomogeneousDiophantine}\left(\left[\left[\exp \left(1\right)\,2^{\frac{1}{2}}\right]\,\left[3^{\frac{1}{3}}\,\mathrm{\pi }\right]\right]\,\left[{10}^{-2}\,{10}^{-4}\right]\right)$

$\left[\left[7484\,\mathrm{-2534}\right]\,\left[16760\,2833\right]\right]$

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

$\left[x=\mathrm{-3050}\,y=\mathrm{-2175}\,z=4450\,r=2093\,v=\mathrm{-13}\right]$

$\mathrm{HomogeneousDiophantine}\left(\left[\left[\log \left(2\right)\,\log \left(5\right)\,3^{\frac{1}{2}}\right]\right]\,\left[{10}^{-2}\right]\,\left[\left[\sin \left(5\right)\,\frac{1}{\log \left(7\right)}\,\exp \left(5\right)\right]\right]\,\left[5\right]\,\left[9\right]\right)$

$\left[\left[\mathrm{-3050}\,\mathrm{-2175}\,4450\right]\,\left[2093\,\mathrm{-13}\right]\right]$

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

For more information on Maple 2016 changes, see Updates in Maple 2016.