Let K be any real valued field, i.e., K is a field together with | | : K → R + such that |xy| = |x||y| and |x + y| ≤ |x| + |y| for all x and y in K, and

1. Introduction

Let K be any real valued field, i.e., K is a field together with | | : K → R ⁺ such that |xy| = |x||y| and |x + y| ≤ |x| + |y| for all x and y in K, and

|z| = 0 ⇔ z = 0.

Let F (X ₁ , X ₂ , · · · , X _n ) = P f _i

_i

···i

X ₁ ⁱ

X ₂ ⁱ

· · · X _n ⁱ

be a formal power series in n indeterminates X 1 , X 2 , · · · , X n (n ≥ 2) with coefficients f i

i

···i

in K, where (K, | |) is a real valued field.

A formal power series F is convergent if there exists a positive real number C such that |f i

i

···i

| ≤ C ⁱ

⁺ⁱ

^+···+i

for all i 1 ≥ 0, i 2 ≥ 0, · · · , i n ≥ 0 and

i ₁ + i ₂ + · · · + i _n 6= 0. If a formal power series F is not convergent, then we say it is divergent.

In [1], S.S. Abhyankar and T.T. M oh proved that the convergent set defined as Conv(F ) = {c ∈ K : F (cX ₂ , X ₂ , · · · , X _n ) is convergent in X ₂ , · · · , X _n } is a set of measure zero if F is divergent in X ₁ , X ₂ , · · · , X _n , where K = R or C.

In [4], A.Sathaye established that the converse of the Abhyankar-Moh’s theorem is also true. i.e., given any small set S, there exists a divergent power series F (X ₁ , X ₂ , · · · , X _n ) with Conv(F ) = S.

In [2],Y.C.Hung generalized the above results to nonlinear restrictions. Given a divergent power series F (X ₁ , X ₂ , · · · , X _n ) in X ₁ , X ₂ , · · · , X _n over K, where K is the field of real or complex numbers, then the convergence sets defined as Conv(F ) = {(c 1 , c 2 , · · · , c m ) ∈ K ^m : F (c 1 X 2 + c 2 X ₂ ² + · · · + c m X ₂ ^m , X 2 , · · · , X n ) is convergent in X ₂ , · · · , X _n } is a measure zero F _σ set.

In this thesis, we shall prove that if F (X ₁ , X ₂ , X ₃ ) is divergent in X ₁ , X ₂ , X ₃ over K where K = R or C, then F (a ¹ X 3 , a 2 X 3 , X 3 ) is divergent for almost all (a ₁ , a ₂ ) ∈ K ² . i.e.,the set

{(a ₁ , a ₂ ) ∈ K ² : F (a ₁ X ₃ , a ₂ X ₃ , X ₃ ) is convergent in X ₃ } is a set of measure zero and is a union of countable number of closed sets.

2