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numtheory

  

kronecker

  

Inhomogeneous Diophantine approximation

 

Calling Sequence

Parameters

Description

Examples

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 x1,x2,,xn,y1,,ym over the integers to a set of inequalities of the form

|a11x1++a1nxnα1y1|err1

..............

|am1x1++amnxnαmym|errm

  

or

valuepa11x1++a1nxnα1y1,p1err1

..............

valuepam1x1++amnxnαmym,pmerrm

• 

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

  

 

x1=...,...,xn=...,y1,...,...,ym=...

  

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 α's are all the same, the list α1,`...`,αm may be replaced by α. 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

withnumtheory:

withpadic:

kronecker0.01log2x+24log5y8312zⅇ2.5u107,3.7ⅇ2x+y+313z513v103,x,y,z,u,v

x=8026,y=3174,z=6916,v=212628,u=218388

(1)

x'x':y'y':u'u':v'v':

kroneckervalueplog5x+log7ylog5u,3320,valuep1xlog7+log11ylog7v,5515,valueplog3x+ⅇ7ylog3w,7712,x,y,u,v,w

x=15516275,y=6404775,w=9747866955,u=1192024656,v=27148890349

(2)

kroneckerlog2,log5,312,ⅇ2,π,313,ⅇ,212,102,105

2863,10057,1494,20761,54906

(3)

kroneckerlog3,log7,log13,sin5,1log7,ⅇ5,2,5,log5,log11,10,15

2000,3125,2825,800,26295606385

(4)

See Also

isolve

numtheory[minkowski]

 


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