# 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 .