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

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)

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

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 = {(.174853 + .626892*ii)x1  + (.350611 + .379495*ii)x1*x2 + (.987173 +
     ------------------------------------------------------------------------
                  2                                           
     .568273*ii)x2  + (.237252 + .116721*ii)x1*x3 + (.184779 +
     ------------------------------------------------------------------------
                                                2             
     .370833*ii)x2*x3 + (.164647 + .713493*ii)x3  + (.444183 +
     ------------------------------------------------------------------------
                                                                
     .644366*ii)x1*x4 + (.339222 + .062212*ii)x2*x4 + (.909537 +
     ------------------------------------------------------------------------
                                                2             
     .566034*ii)x3*x4 + (.562839 + .629991*ii)x4  + (.194945 +
     ------------------------------------------------------------------------
                                                               
     .518585*ii)x1*x5 + (.465736 + .40273*ii)x2*x5 + (.305423 +
     ------------------------------------------------------------------------
                                                               
     .732358*ii)x3*x5 + (.479826 + .815167*ii)x4*x5 + (.97723 +
     ------------------------------------------------------------------------
                   2                                                        
     .0595849*ii)x5  + (.914199 + .887381*ii)x1 + (.169813 + .965004*ii)x2 +
     ------------------------------------------------------------------------
                                                                       
     (.0647412 + .877846*ii)x3 + (.0340514 + .507989*ii)x4 + (.150294 +
     ------------------------------------------------------------------------
                                               2             
     .656391*ii)x5 - 1, (.305946 + .53632*ii)x1  + (.205375 +
     ------------------------------------------------------------------------
                                                2              
     .276652*ii)x1*x2 + (.280679 + .475179*ii)x2  + (.0958269 +
     ------------------------------------------------------------------------
                                                                
     .605398*ii)x1*x3 + (.130004 + .522285*ii)x2*x3 + (.767433 +
     ------------------------------------------------------------------------
                  2                                           
     .155346*ii)x3  + (.883549 + .942865*ii)x1*x4 + (.171029 +
     ------------------------------------------------------------------------
                                                                  
     .340019*ii)x2*x4 + (.0215389 + .283851*ii)x3*x4 + (.0328338 +
     ------------------------------------------------------------------------
                 2                                           
     .21113*ii)x4  + (.0821679 + .10394*ii)x1*x5 + (.818142 +
     ------------------------------------------------------------------------
                                                                
     .781168*ii)x2*x5 + (.971588 + .249992*ii)x3*x5 + (.611415 +
     ------------------------------------------------------------------------
                                                  2              
     .638389*ii)x4*x5 + (.0594513 + .0625324*ii)x5  + (.0645275 +
     ------------------------------------------------------------------------
                                                                          
     .283709*ii)x1 + (.977573 + .212436*ii)x2 + (.592747 + .831802*ii)x3 +
     ------------------------------------------------------------------------
                                                                       
     (.501243 + .154289*ii)x4 + (.467203 + .765564*ii)x5 - 1, (.46452 +
     ------------------------------------------------------------------------
                 2                                             
     .16645*ii)x1  + (.224737 + .00415468*ii)x1*x2 + (.905002 +
     ------------------------------------------------------------------------
                  2                                           
     .502056*ii)x2  + (.370963 + .511938*ii)x1*x3 + (.888883 +
     ------------------------------------------------------------------------
                                                2             
     .0776828*ii)x2*x3 + (.35911 + .751187*ii)x3  + (.806831 +
     ------------------------------------------------------------------------
                                                               
     .532303*ii)x1*x4 + (.41855 + .138169*ii)x2*x4 + (.252537 +
     ------------------------------------------------------------------------
                                               2             
     .10246*ii)x3*x4 + (.475683 + .107149*ii)x4  + (.572433 +
     ------------------------------------------------------------------------
                                                                
     .269298*ii)x1*x5 + (.543508 + .212989*ii)x2*x5 + (.543294 +
     ------------------------------------------------------------------------
                                                               
     .659792*ii)x3*x5 + (.13359 + .420228*ii)x4*x5 + (.112527 +
     ------------------------------------------------------------------------
                  2                                                        
     .659052*ii)x5  + (.270725 + .075503*ii)x1 + (.581271 + .640172*ii)x2 +
     ------------------------------------------------------------------------
                                                                     
     (.919548 + .560684*ii)x3 + (.847715 + .763255*ii)x4 + (.477291 +
     ------------------------------------------------------------------------
                                                 2              
     .0048212*ii)x5 - 1, (.689783 + .231832*ii)x1  + (.0311856 +
     ------------------------------------------------------------------------
                                                2              
     .388518*ii)x1*x2 + (.762468 + .455803*ii)x2  + (.0258757 +
     ------------------------------------------------------------------------
                                                                
     .0288963*ii)x1*x3 + (.299391 + .77016*ii)x2*x3 + (.429347 +
     ------------------------------------------------------------------------
                  2                                           
     .288364*ii)x3  + (.912808 + .898905*ii)x1*x4 + (.885699 +
     ------------------------------------------------------------------------
                                                                 
     .761573*ii)x2*x4 + (.993594 + .895324*ii)x3*x4 + (.0770766 +
     ------------------------------------------------------------------------
                 2                                                           
     .30813*ii)x4  + (.580909 + .228196*ii)x1*x5 + (.42526 + .403967*ii)x2*x5
     ------------------------------------------------------------------------
                                                                            
     + (.34672 + .363895*ii)x3*x5 + (.633138 + .262856*ii)x4*x5 + (.559948 +
     ------------------------------------------------------------------------
                  2                                                          
     .249212*ii)x5  + (.649378 + .564001*ii)x1 + (.455949 + .00471457*ii)x2 +
     ------------------------------------------------------------------------
                                                                   
     (.771746 + .74002*ii)x3 + (.20217 + .774737*ii)x4 + (.556033 +
     ------------------------------------------------------------------------
                                               2             
     .91523*ii)x5 - 1, (.531857 + .356922*ii)x1  + (.172651 +
     ------------------------------------------------------------------------
                                                 2              
     .533417*ii)x1*x2 + (.738044 + .0215483*ii)x2  + (.0219272 +
     ------------------------------------------------------------------------
                                                                
     .817573*ii)x1*x3 + (.751374 + .767024*ii)x2*x3 + (.886882 +
     ------------------------------------------------------------------------
                  2                                           
     .784812*ii)x3  + (.168645 + .358533*ii)x1*x4 + (.553621 +
     ------------------------------------------------------------------------
                                                               
     .74003*ii)x2*x4 + (.829966 + .270696*ii)x3*x4 + (.831694 +
     ------------------------------------------------------------------------
                  2                                           
     .729951*ii)x4  + (.255456 + .466765*ii)x1*x5 + (.545222 +
     ------------------------------------------------------------------------
                                                               
     .958457*ii)x2*x5 + (.790306 + .783351*ii)x3*x5 + (.29683 +
     ------------------------------------------------------------------------
                                                2
     .162247*ii)x4*x5 + (.908964 + .231486*ii)x5  + (.671836 + .707257*ii)x1
     ------------------------------------------------------------------------
     + (.0423322 + .839258*ii)x2 + (.821811 + .68646*ii)x3 + (.101507 +
     ------------------------------------------------------------------------
     .253324*ii)x4 + (.223416 + .0263964*ii)x5 - 1}

o2 : List

Ways to use randomSystem:

randomSystem(ZZ,ZZ,Ring)

For the programmer

The object randomSystem is a method function.

The source of this document is in ExampleSystems/randomSystem.m2:41:0.