By H. Piaggio
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Extra info for An Elementary Treatise on Diff. Eqns. and Their Applns.
Fn ) and I = f1 , . . , fn ⊆ C[x1 , . . , xn ]. A solution p ∈ V (I) is called singular if the Jacobi matrix JF (p) is singular. On the other hand, if JF (p) is invertible, then p is called non-singular or regular. 4). We present another result in this direction. 3 Let f1 , . . , fn ∈ C[x1 , . . , xn ], I = f1 , . . , fn and F = (f1 , . . , fn ). If for some point x ∈ V (I) we have det(JF (x)) = 0, then x is isolated. Proof. According to the inverse function theorem there is an open set X around x which via F is in bijection with a set Y .
Xn ]Mj . This is the fourth statement. For any i: ei (pi ) = 1 ⇒ e1i ∈ C[x1 , . . , xn ]Mi ⇒ ei − 1 = e1i (ei (ei − 1)) ∈ IC[x1 , . . , xn ]Mi . This is the fifth statement. Let g ∈ C[x1 , . . , xn ] \ Mi . Then g(pi ) = 0. Without loss of generality we may assume g(pi ) = 1, implying 1−g ∈ Mi . Choose h = ei (1+(1−g)+· · ·+(1− g)d−1 ). Then hg−ei = h(1−(1−g))−ei = ei (1+· · ·+(1−g)d−1 )(1−(1−g))−ei = ei (1 + (1 − g)d ) − ei = ei (1 − g)d . Observation 1 says that ei ∈ Mjd for j = i. Furthermore (1 − g)d ∈ Mid .
Gs } is a Gr¨obner basis we can write every S-polynomial as S≺ (gi , gj ) = k ak fk with fk ∈ G, ak a monomial in C(t)[x1 , . . , xn ] and in≺ (ak fk ) in≺ (gi , gj ). ) Do this for all finitely many choices of i and j. In such expressions some coefficients from the field C(t) will show up in the expression and the expanded expression. Maybe these cannot be evaluated for all choices of t. Let P be the product of all the denominators and numerators of all the coefficients. If t ∈ C satisfies P (t ) = 0 then we may substitute the value t for the variable t in our expressions.
An Elementary Treatise on Diff. Eqns. and Their Applns. by H. Piaggio