PREFACE
Our knowledg e abou t fractiona l part s o f linea r polynomial s i s fairl y satisfactory .
Dirichlet's Theore m tell s u s tha t fo r ever y a an d fo r N 1, ther e i s a natura l n N
with ||cc72I I N" , wher e | | || denote s th e distanc e t o th e neares t integer ; an d thi s
bound i s bes t possible . Ou r knowledge abou t fractiona l part s o f nonlinea r polynomial
is no t s o satisfactory . I n these Note s w e star t ou t wit h Heilbronn' s Theore m o n quad -
ratic polynomial s fin) = an , accordin g t o whic h ther e i s a natura l n N wit h ||/(w)| |
N~ " K. Fro m thi s w e branc h ou t i n thre e directions . I n §§7—12 w e dea l wit h arbi
trary polynomial s wit h constan t ter m zero . I n §§13—19 w e tak e u p simultaneou s approx
imation o f quadrati c polynomials , an d i n §§20 , 2 1 w e discus s specia l quadrati c poly -
nomials i n severa l variables . Ther e ar e man y ope n questions ; i n fact , mos t o f th e re -
sults obtaine d i n thes e Note s ar e almos t certainl y no t bes t possible . Sinc e th e theor y
is no t i n it s fina l form , I have refraine d fro m includin g th e mos t genera l situation , i.e
simultaneous fractiona l part s o f polynomial s i n severa l variable s o f arbitrar y degree .
On th e othe r hand , I hav e give n al l th e proof s i n ful l detail , a t a leisurel y pace .
I wis h t o than k th e Nationa l Scienc e Foundatio n an d th e Illinoi s Stat e Universit y
for sponsorin g thi s serie s o f lectures .
Wolfgang M . Schmid t
December 1976
For furthe r references , coverin g a rathe r wide r area , se e Malyshe v an d Podsypani n [197
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