next | previous | forward | backward | up | top | index | toc | Macaulay2 website
ExampleSystems :: randomSystem

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

Synopsis

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           3   3  2     5     2   6  3   1  2    7       
o1 = {-x1  + -x1 x2 + -x1*x2  + -x2  + -x1  + -x1*x2 + x2  + -x1 + -x2 - 1, 7x1  + -x1 x2 + -x1*x2  + -x2  + -x1  + --x1*x2 +
      4      2        4         4      4      2              2     2               7        2         7      2      10       
     ----------------------------------------------------------------------------------------------------------------------------
     7  2   7      7
     -x2  + -x1 + --x2 - 1}
     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                                                           2                               
o2 = {(.602256 + .860413*ii)x1  + (.0153057 + .871288*ii)x1*x2 + (.0741364 + .701252*ii)x2  + (.137183 + .33307*ii)x1*x3 +
     ----------------------------------------------------------------------------------------------------------------------------
                                                          2                                                              
     (.433705 + .1996*ii)x2*x3 + (.0819502 + .985468*ii)x3  + (.816738 + .359003*ii)x1*x4 + (.332797 + .575146*ii)x2*x4 +
     ----------------------------------------------------------------------------------------------------------------------------
                                                          2                                                                
     (.031794 + .324112*ii)x3*x4 + (.138528 + .74322*ii)x4  + (.721791 + .358442*ii)x1*x5 + (.254327 + .00503878*ii)x2*x5 +
     ----------------------------------------------------------------------------------------------------------------------------
                                                                                         2                                       
     (.542391 + .606624*ii)x3*x5 + (.318163 + .396733*ii)x4*x5 + (.592648 + .520144*ii)x5  + (.0430999 + .500492*ii)x1 + (.387244
     ----------------------------------------------------------------------------------------------------------------------------
                                                                                                                   
     + .98071*ii)x2 + (.65684 + .215998*ii)x3 + (.810424 + .955156*ii)x4 + (.618011 + .124714*ii)x5 - 1, (.241927 +
     ----------------------------------------------------------------------------------------------------------------------------
                  2                                                        2                                           
     .396394*ii)x1  + (.472065 + .744753*ii)x1*x2 + (.15219 + .405122*ii)x2  + (.975415 + .743894*ii)x1*x3 + (.601959 +
     ----------------------------------------------------------------------------------------------------------------------------
                                              2                                                                           
     .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 +
     ----------------------------------------------------------------------------------------------------------------------------
                                                                              2                                        
     .92149*ii)x3*x5 + (.247135 + .831517*ii)x4*x5 + (.450832 + .0985344*ii)x5  + (.687018 + .908397*ii)x1 + (.177961 +
     ----------------------------------------------------------------------------------------------------------------------------
                                                                                                                             2  
     .624105*ii)x2 + (.63835 + .687791*ii)x3 + (.142063 + .249642*ii)x4 + (.352096 + .8528*ii)x5 - 1, (.828915 + .45455*ii)x1  +
     ----------------------------------------------------------------------------------------------------------------------------
                                                           2                                                                     
     (.665644 + .304074*ii)x1*x2 + (.561231 + .771332*ii)x2  + (.74293 + .933838*ii)x1*x3 + (.60143 + .217061*ii)x2*x3 + (.898074
     ----------------------------------------------------------------------------------------------------------------------------
                   2                                                                                                       
     + .33915*ii)x3  + (.874974 + .952733*ii)x1*x4 + (.211264 + .417718*ii)x2*x4 + (.694798 + .796494*ii)x3*x4 + (.284861 +
     ----------------------------------------------------------------------------------------------------------------------------
                  2                                                                                                        
     .515519*ii)x4  + (.210005 + .966078*ii)x1*x5 + (.711184 + .366998*ii)x2*x5 + (.0324724 + .980272*ii)x3*x5 + (.685796 +
     ----------------------------------------------------------------------------------------------------------------------------
                                                2                                                                   
     .151442*ii)x4*x5 + (.361682 + .988025*ii)x5  + (.160149 + .477978*ii)x1 + (.447057 + .382018*ii)x2 + (.349223 +
     ----------------------------------------------------------------------------------------------------------------------------
                                                                                                      2             
     .138181*ii)x3 + (.72901 + .367262*ii)x4 + (.0974663 + .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 +
     ----------------------------------------------------------------------------------------------------------------------------
                                                                                                       2             
     .178198*ii)x3 + (.0751062 + .330319*ii)x4 + (.665784 + .568307*ii)x5 - 1, (.765445 + .193804*ii)x1  + (.924856 +
     ----------------------------------------------------------------------------------------------------------------------------
                                                2                                                                         
     .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 :

For the programmer

The object randomSystem is a method function.