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NumberTheory

  

HomogeneousDiophantine

  

solution to Minkowski's linear forms

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

HomogeneousDiophantine(ineqs, xvars, yvars)

HomogeneousDiophantine(real_cfs, real_errors)

HomogeneousDiophantine(padic_cfs, adicities, padic_errors)

HomogeneousDiophantine(real_cfs, real_errors, padic_cfs, adicities, padic_errors)

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 x1,,xn,y1,,ym over the integers to a set of inequalities of the form

a1,1x1+a1,nxn+...y1err1

...

aj,1x1+aj,nxn+...yjerrj

  

or   

padic:−valuepaj+1,1x1+aj+1,nxn+...yj+1,pj+1pj+1errj+1

...

padic:−valuepam,1x1+am,nxn+...ym,pmpmerrm

  

where padic:−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

x1=s1,...,xn=sn,y1=t1,...,ym=tm

• 

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,

s1,...,sn,t1,...,tm

Examples

withNumberTheory:

HomogeneousDiophantineabssqrt2xy103,x,y

x=5741,y=8119

(1)

withpadic:

HomogeneousDiophantineabs313z1+πz2s2104,absexp1z1+212z2s1102,z1,z2,s1,s2

z1=1271,z2=6970,s2=23730,s1=13312

(2)

An equivalent matrix form calling sequence is:

HomogeneousDiophantineexp1,212,313,π,102,104

7484,−2534,16760,2833

(3)

The solutions may be different but both are valid.

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

HomogeneousDiophantineabslog2x+log5y+312zr102,valuepsin5x+1log7y+exp5zv,559,x,y,z,r,v

x=−3050,y=−2175,z=4450,r=2093,v=−13

(4)

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

HomogeneousDiophantinelog2,log5,312,102,sin5,1log7,exp5,5,9

−3050,−2175,4450,2093,−13

(5)

Compatibility

• 

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

• 

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

See Also

isolve

NumberTheory

NumberTheory[InhomogeneousDiophantine]

padic[valuep]