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Homework Statement
I am reading the undergraduate introduction to algebraic geometry entitled "Ideals, Varieties and Algorithms: An introduction to Computational Algebraic Geometry and Commutative Algebra (Third Edition) by David Cox, John Little and Donal O'Shea ... ...
I am currently focused on Chapter 4, Section 1: Hilbert's Nullstellensatz ... ... and need help with the aspects of Cox et al's interesting proof of the Weak Nullstellensatz as outlined in Exercise 3 ...
Exercise 3 (Chapter 4, Section 1) reads as follows:
As Exercise 3 refers to aspects of the proof of Theorem 1: The Weak Nullstellensatz, I am providing the first part of the proof of that Theorem as follows:
My question is as follows:
How do we formulate and state a formal and rigorous proof of 3(a) ... ... that is a proof of the proposition that the coefficient ##c(a_1, a_2, \ ... \ ... \ , a_n )## of ##\tilde{x}^N_1## in ##f## is ##h_N (1, a_2, \ ... \ ... \ , a_n )## ... ...
Homework Equations
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Relevant equations (such as the transformation from ##f_1## to ##\tilde{f}_1## are included in the attempt at a solution ...
The Attempt at a Solution
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Being unable to formulate a rigorous and general proof I have studies an example case which indicates the proposition is likely to be true ... but not sure about formulating a general proof ... the worked example follows ...
Consider ##f_1 = 3 x_1^2 x_2^2 x_3 + 2 x_2^2 x_3^2##
The total degree of ##f_1## is ##N= 5##, determined by the first term, namely ##3 x_1^2 x_2^2 x_3## ... ...
Also note that ##h_N(x_1, x_2, x_3) = h_5(x_1, x_2, x_3) = 3 x_1^2 x_2^2 x_3##
and that ##h_5( 1, a_2, a_3) = 3 a_2^2 a_3##
... ... ...
Consider now the transformation f_1 \mapsto \tilde{f_1} given by:
##x_1 = \tilde{x}_1##
##x_2 = \tilde{x}_2 + a_2 \tilde{x}_1##
... ...
... ...
##x_n = \tilde{x}_n + a_n \tilde{x}_1##
Considering the above transformation, it is clear that the term ##3 x_1^2 x_2^2 x_3## will give rise the the coefficient of## \tilde{x}^N_1## ... ... so we examine this term under the transformation ... ...
... so then ...
##3 x_1^2 x_2^2 x_3 = 3 \tilde{x}_1^2 ( \tilde{x}_2^2 + 2 a_2 \tilde{x}_1 \tilde{x}_2 + a_2^2 \tilde{x}_1^2 ) ( \tilde{x}_3 + a_3 \tilde{x}_1 )##
##= ( 3 \tilde{x}_1^2 \tilde{x}_2^2 + 6 a_2 \tilde{x}_1^3 \tilde{x}_2 + 3 a_2^2 \tilde{x}_1^4 ) ( \tilde{x}_3 + a_3 \tilde{x}_1 )##
Clearly, the term involving ##\tilde{x}_1^N = \tilde{x}_1^5## will be
##3 a_2^2 a_3 \tilde{x}_1^5##
So we have that ##h_N( \tilde{x}_1, \tilde{x}_2, \tilde{x}_3 )## in ##\tilde{f}_1## is
##h_5( \tilde{x}_1, \tilde{x}_2, \tilde{x}_3 ) = 3 a_2^2 a_3 \tilde{x}_1^5##
so we have that
##h_5( 1, a_2, a_3 ) = 3 a_2^2 a_3##
as required ... ... BUT ...
... ... how do we formulate and state a formal proof of the general proposition that the coefficient ##c(a_1, a_2, \ ... \ ... \ , a_n )## of ## \tilde{x}^N_1## in ##f## is ##h_N (1, a_2, \ ... \ ... \ , a_n )## ... ... ?
Does the proof just describe the computation process in general terms ... ?
Hope someone can help ... ...
Help will be appreciated ...
Peter
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