▁as ▁the ▁order ▁of ▁ ▁contact ▁( or ▁local ▁intersection ▁number ) ▁of ▁the ▁graph ▁y ▁= ▁f ( x ) ▁with ▁the ▁x - axis ▁y ▁= ▁ 0 ▁near ▁the ▁point ▁( a , 0 ). ▁If , ▁instead ▁of ▁the ▁x - axis , ▁one ▁fixes ▁another ▁ir reducible ▁plane ▁curve ▁h ( x , y ) ▁= ▁ 0 ▁and ▁a ▁point ▁( a , b ), ▁one ▁may ▁similarly ▁define ▁a ▁valu ation ▁v h ▁on ▁ ▁so ▁that ▁v h ( f ) ▁is ▁the ▁order ▁of ▁contact ▁( the ▁intersection ▁number ) ▁between ▁the ▁fixed ▁curve ▁and ▁f ( x , y ) ▁= ▁ 0 ▁near ▁( a , b ). ▁This ▁valu ation ▁naturally ▁extends ▁to ▁rational ▁functions ▁▁ ▁In ▁fact , ▁this ▁construction ▁is ▁a ▁special ▁case ▁of ▁the ▁ π - adic ▁valu ation ▁on ▁a ▁P ID ▁defined ▁above . ▁Nam ely , ▁consider ▁the ▁local ▁ring ▁, ▁the ▁ring ▁of ▁rational ▁functions ▁which ▁are ▁defined ▁on ▁some ▁open ▁subset ▁of ▁the ▁curve ▁h ▁= ▁ 0 . ▁This ▁is ▁a ▁P ID ; ▁in ▁fact ▁a ▁discrete ▁valu ation ▁ring ▁whose ▁only ▁ide als ▁are ▁the ▁powers ▁. ▁Then ▁the ▁above ▁valu ation ▁v h ▁is ▁the ▁ π - adic ▁valu ation ▁corresponding ▁to ▁the ▁ir reducible ▁element ▁ π ▁= ▁h ▁ ∈ ▁R . ▁ ▁Example : ▁Consider ▁the ▁curve ▁ ▁defined ▁by ▁, ▁namely ▁the ▁graph ▁ ▁near ▁the ▁origin ▁. ▁This ▁curve ▁can ▁be ▁paramet