randomSystem -- an example of a 0-dimensional square polynomial system

Synopsis

Usage:

F = randomSystem(n,d,K)
Inputs:
- n, an integer, positive value representing the number of polynomials
- d, an integer, positive value representing the degree of the polynomials
- K, a ring, usually a field, e.g., QQ or CC_{53}
Outputs:
- F, a list, a system of $n$ polynomials random polynomials of degree $d$ in $n$ variables.

Description

This system was solved in May 2020, using track in Macaulay2 v1.15 with an Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.

For a system with 2 polynomials of degree 3 over rationals there were 9 solutions found in 0.06 seconds.

For a system with 5 polynomials of degree 2 over complex numbers there were 32 solutions found in 0.297 seconds.

i1 : randomSystem(2,3,QQ)

      3  3   3  2     3     2   7  3   9  2   1          2   9     1       
o1 = {-x1  + -x1 x2 + -x1*x2  + -x2  + -x1  + -x1*x2 + x2  + -x1 + -x2 - 1,
      4      2        4         4      4      2              2     2       
     ------------------------------------------------------------------------
        3   3  2     5     2   6  3   1  2    7        7  2   7      7
     7x1  + -x1 x2 + -x1*x2  + -x2  + -x1  + --x1*x2 + -x2  + -x1 + --x2 - 1}
            7        2         7      2      10        3      9     10

o1 : List

i2 : randomSystem(5, 2, CC_53)
-- warning: experimental computation over inexact field begun
--          results not reliable (one warning given per session)

                              2                                             
o2 = {(.602256 + .860413*ii)x1  + (.0153057 + .871288*ii)x1*x2 + (.0741364 +
     ------------------------------------------------------------------------
                  2                                                         
     .701252*ii)x2  + (.137183 + .33307*ii)x1*x3 + (.433705 + .1996*ii)x2*x3
     ------------------------------------------------------------------------
                                2                                           
     + (.0819502 + .985468*ii)x3  + (.816738 + .359003*ii)x1*x4 + (.332797 +
     ------------------------------------------------------------------------
                                                                
     .575146*ii)x2*x4 + (.031794 + .324112*ii)x3*x4 + (.138528 +
     ------------------------------------------------------------------------
                 2                                           
     .74322*ii)x4  + (.721791 + .358442*ii)x1*x5 + (.254327 +
     ------------------------------------------------------------------------
                                                                  
     .00503878*ii)x2*x5 + (.542391 + .606624*ii)x3*x5 + (.318163 +
     ------------------------------------------------------------------------
                                                2                            
     .396733*ii)x4*x5 + (.592648 + .520144*ii)x5  + (.0430999 + .500492*ii)x1
     ------------------------------------------------------------------------
                                                                     
     + (.387244 + .98071*ii)x2 + (.65684 + .215998*ii)x3 + (.810424 +
     ------------------------------------------------------------------------
                                                                           2
     .955156*ii)x4 + (.618011 + .124714*ii)x5 - 1, (.241927 + .396394*ii)x1 
     ------------------------------------------------------------------------
                                                            2             
     + (.472065 + .744753*ii)x1*x2 + (.15219 + .405122*ii)x2  + (.975415 +
     ------------------------------------------------------------------------
                                                                            2
     .743894*ii)x1*x3 + (.601959 + .23788*ii)x2*x3 + (.86251 + .144245*ii)x3 
     ------------------------------------------------------------------------
                                                                             
     + (.583216 + .0680562*ii)x1*x4 + (.968104 + .0260925*ii)x2*x4 + (.993596
     ------------------------------------------------------------------------
                                                  2              
     + .516494*ii)x3*x4 + (.370634 + .973687*ii)x4  + (.0867182 +
     ------------------------------------------------------------------------
                                                                
     .327113*ii)x1*x5 + (.504847 + .80403*ii)x2*x5 + (.0460269 +
     ------------------------------------------------------------------------
                                                               
     .92149*ii)x3*x5 + (.247135 + .831517*ii)x4*x5 + (.450832 +
     ------------------------------------------------------------------------
                   2                                                        
     .0985344*ii)x5  + (.687018 + .908397*ii)x1 + (.177961 + .624105*ii)x2 +
     ------------------------------------------------------------------------
                                                                    
     (.63835 + .687791*ii)x3 + (.142063 + .249642*ii)x4 + (.352096 +
     ------------------------------------------------------------------------
                                             2                              
     .8528*ii)x5 - 1, (.828915 + .45455*ii)x1  + (.665644 + .304074*ii)x1*x2
     ------------------------------------------------------------------------
                               2                                         
     + (.561231 + .771332*ii)x2  + (.74293 + .933838*ii)x1*x3 + (.60143 +
     ------------------------------------------------------------------------
                                               2             
     .217061*ii)x2*x3 + (.898074 + .33915*ii)x3  + (.874974 +
     ------------------------------------------------------------------------
                                                                
     .952733*ii)x1*x4 + (.211264 + .417718*ii)x2*x4 + (.694798 +
     ------------------------------------------------------------------------
                                                2             
     .796494*ii)x3*x4 + (.284861 + .515519*ii)x4  + (.210005 +
     ------------------------------------------------------------------------
                                                                 
     .966078*ii)x1*x5 + (.711184 + .366998*ii)x2*x5 + (.0324724 +
     ------------------------------------------------------------------------
                                                                
     .980272*ii)x3*x5 + (.685796 + .151442*ii)x4*x5 + (.361682 +
     ------------------------------------------------------------------------
                  2                                                        
     .988025*ii)x5  + (.160149 + .477978*ii)x1 + (.447057 + .382018*ii)x2 +
     ------------------------------------------------------------------------
                                                                     
     (.349223 + .138181*ii)x3 + (.72901 + .367262*ii)x4 + (.0974663 +
     ------------------------------------------------------------------------
                                                2             
     .764008*ii)x5 - 1, (.759969 + .403568*ii)x1  + (.495812 +
     ------------------------------------------------------------------------
                                               2             
     .280304*ii)x1*x2 + (.30991 + .582056*ii)x2  + (.996962 +
     ------------------------------------------------------------------------
                                                                
     .396245*ii)x1*x3 + (.450918 + .541212*ii)x2*x3 + (.831293 +
     ------------------------------------------------------------------------
                 2                                            
     .55344*ii)x3  + (.831996 + .832458*ii)x1*x4 + (.0931882 +
     ------------------------------------------------------------------------
                                                                
     .418325*ii)x2*x4 + (.983385 + .115868*ii)x3*x4 + (.449411 +
     ------------------------------------------------------------------------
                  2                                           
     .650608*ii)x4  + (.905355 + .553319*ii)x1*x5 + (.433248 +
     ------------------------------------------------------------------------
                                                                
     .460527*ii)x2*x5 + (.236088 + .895839*ii)x3*x5 + (.287687 +
     ------------------------------------------------------------------------
                                                 2              
     .906977*ii)x4*x5 + (.0745177 + .339464*ii)x5  + (.0323952 +
     ------------------------------------------------------------------------
                                                                          
     .994173*ii)x1 + (.804288 + .611552*ii)x2 + (.161346 + .178198*ii)x3 +
     ------------------------------------------------------------------------
                                                                         
     (.0751062 + .330319*ii)x4 + (.665784 + .568307*ii)x5 - 1, (.765445 +
     ------------------------------------------------------------------------
                  2                                                         2
     .193804*ii)x1  + (.924856 + .817228*ii)x1*x2 + (.317573 + .793382*ii)x2 
     ------------------------------------------------------------------------
                                                                             
     + (.250585 + .820297*ii)x1*x3 + (.726918 + .680856*ii)x2*x3 + (.321441 +
     ------------------------------------------------------------------------
                  2                                           
     .996312*ii)x3  + (.571536 + .539913*ii)x1*x4 + (.200531 +
     ------------------------------------------------------------------------
                                                                 
     .404241*ii)x2*x4 + (.0585566 + .744873*ii)x3*x4 + (.695115 +
     ------------------------------------------------------------------------
                  2                                           
     .142609*ii)x4  + (.295863 + .823467*ii)x1*x5 + (.184098 +
     ------------------------------------------------------------------------
                                                                 
     .481035*ii)x2*x5 + (.934629 + .705903*ii)x3*x5 + (.0415199 +
     ------------------------------------------------------------------------
                                                 2
     .150125*ii)x4*x5 + (.0231209 + .623309*ii)x5  + (.195174 +
     ------------------------------------------------------------------------
     .0321649*ii)x1 + (.546024 + .0872153*ii)x2 + (.651965 + .754163*ii)x3 +
     ------------------------------------------------------------------------
     (.159008 + .0823846*ii)x4 + (.545843 + .0533627*ii)x5 - 1}

o2 : List

Ways to use randomSystem :

randomSystem(ZZ,ZZ,Ring)

For the programmer

The object randomSystem is a method function.