s ▁and ▁the ▁rank ▁of ▁the ▁matrix ▁of ▁the ▁system . ▁ ▁Und erd et erm ined ▁polynomial ▁systems ▁ ▁The ▁main ▁property ▁of ▁linear ▁under det erm ined ▁systems , ▁of ▁having ▁either ▁no ▁solution ▁or ▁infinitely ▁many , ▁extends ▁to ▁systems ▁of ▁polynomial ▁equations ▁in ▁the ▁following ▁way . ▁ ▁A ▁system ▁of ▁polynomial ▁equations ▁which ▁has ▁fewer ▁equations ▁than ▁unknown s ▁is ▁said ▁to ▁be ▁under det erm ined . ▁It ▁has ▁either ▁infinitely ▁many ▁complex ▁solutions ▁( or , ▁more ▁generally , ▁solutions ▁in ▁an ▁algebra ically ▁closed ▁field ) ▁or ▁is ▁incons istent . ▁It ▁is ▁incons istent ▁if ▁and ▁only ▁if ▁ ▁is ▁a ▁linear ▁combination ▁( with ▁polynomial ▁coefficients ) ▁of ▁the ▁equations ▁( this ▁is ▁Hil bert ' s ▁Null stell ens atz ). ▁If ▁an ▁under det erm ined ▁system ▁of ▁t ▁equations ▁in ▁n ▁variables ▁( t ▁< ▁n ) ▁has ▁solutions , ▁then ▁the ▁set ▁of ▁all ▁complex ▁solutions ▁is ▁an ▁algebraic ▁set ▁of ▁dimension ▁at ▁least ▁. ▁If ▁the ▁under det erm ined ▁system ▁is ▁chosen ▁at ▁random ▁the ▁dimension ▁is ▁equal ▁to ▁ ▁with ▁probability ▁one . ▁ ▁Und erd et erm ined ▁systems ▁with ▁other ▁constraints ▁and ▁in ▁optimization ▁problems ▁ ▁In ▁general , ▁an ▁under det erm ined ▁system ▁of ▁linear ▁equations ▁has ▁an ▁infinite ▁number ▁of ▁solutions , ▁if ▁any . ▁However , ▁in ▁ ▁optimization ▁problems ▁that ▁are ▁subject ▁to ▁linear ▁equality ▁constraints , ▁only ▁one ▁of ▁the ▁solutions ▁is ▁relevant , ▁namely ▁the ▁one ▁giving