▁of ▁such ▁fields ▁are ▁function ▁fields ▁( or ▁power ▁series ▁fields ) ▁of ▁one ▁variable ▁over ▁perfect ▁base ▁fields . ▁ ▁Qu adr atic ▁forms ▁over ▁F 2 ▁ ▁Over ▁F 2 , ▁the ▁Ar f ▁invariant ▁is ▁ 0 ▁if ▁the ▁quadratic ▁form ▁is ▁equivalent ▁to ▁a ▁direct ▁sum ▁of ▁copies ▁of ▁the ▁binary ▁form ▁, ▁and ▁it ▁is ▁ 1 ▁if ▁the ▁form ▁is ▁a ▁direct ▁sum ▁of ▁ ▁with ▁a ▁number ▁of ▁copies ▁of ▁. ▁ ▁William ▁Brow der ▁has ▁called ▁the ▁Ar f ▁invariant ▁the ▁dem ocr atic ▁invariant ▁because ▁it ▁is ▁the ▁value ▁which ▁is ▁assumed ▁most ▁often ▁by ▁the ▁quadratic ▁form . ▁Another ▁character ization : ▁q ▁has ▁Ar f ▁invariant ▁ 0 ▁if ▁and ▁only ▁if ▁the ▁underlying ▁ 2 k - dimensional ▁vector ▁space ▁over ▁the ▁field ▁F 2 ▁has ▁a ▁k - dimensional ▁sub space ▁on ▁which ▁q ▁is ▁ident ically ▁ 0 ▁– ▁that ▁is , ▁a ▁totally ▁is otrop ic ▁sub space ▁of ▁half ▁the ▁dimension . ▁In ▁other ▁words , ▁a ▁n ons ing ular ▁quadratic ▁form ▁of ▁dimension ▁ 2 k ▁has ▁Ar f ▁invariant ▁ 0 ▁if ▁and ▁only ▁if ▁its ▁is ot ropy ▁index ▁is ▁k ▁( this ▁is ▁the ▁maximum ▁dimension ▁of ▁a ▁totally ▁is otrop ic ▁sub space ▁of ▁a ▁n ons ing ular ▁form ). ▁ ▁The ▁Ar f ▁invariant ▁in ▁topology ▁ ▁Let ▁M ▁be ▁a ▁compact , ▁connected ▁ 2 k - dimensional ▁manifold ▁with ▁a ▁boundary ▁ ▁such ▁that ▁the ▁induced ▁morph