$\newcommand{\kk}{\mathbb{K}}$
This is the first of a number, hopefully greater than one, of posts in which I record my understanding of Étienne Bézout’s approach towards his theorem on number of (isolated) solutions of (square) polynomial systems. My source is Eric Feron’s translation (General Theory of Algebraic Equations, Princeton University Press, 2006) of Bézout’s Théorie Générale des Équations Algébriques published in 1779.
As far as I understand from reading Section 1 of Book one (up to Paragraph 47, where the theorem is “proved”), Bézout tries to prove the following statement: “the number of solutions (over an algebraically closed fied) of $n$ generic polynomials in $n$ variables is the product of their degrees“; however, his proof is not complete. The following is what I understood.
- Let $f_1, \ldots, f_n$ be polynomials in indeterminates $(x_1, \ldots, x_n)$ over an algebraically closed field $\kk$.
- Bézout claims (without proof) the number of solutions of $f_1, \ldots, f_n$ should be the same as the smallest possible degree of a univariate polynomial in $x_1$ (presumably some version of the resultant) obtained from $f_1$ by applying the “substitutions” implied by the relations $f_i = 0$, $i = 2, \ldots, n$.
- In general the optimal number of “substitutions” are possible only after multiplying $f_1$ by polynomials (in $x_1, \ldots, x_n$) of sufficiently high degree; these polynomials are called “polynomial multipliers”.
- Write $d_i := \deg(f_i)$, $i = 1, \ldots, n$, and let $d \gg 1$ be the degree of the polynomial multiplier. Treating the equations $f_ig = 0$ (where $g$ is a polynomial) as linear conditions on coefficients, he considers the quotient of the vector space $P_{d+d_1}$ of polynomials of degree $\leq d+d_1$ modulo the subspace $\sum_{i=2}^n f_iP_{d+d_1 – d_i}$. He computes the dimension of this space in the case that $f_i = x_i^{d_i}$, $i = 2, \ldots, n$, and (implicitly) claims without proof that this would continue to be true if $f_i$ are generic “complete” polynomials of degree $d_i$ (here “complete” means every monomial of degree $<= d_i$ is present in $f_i$).
- Similarly, he computes the dimension of $P_d/\sum_{i=2}^n f_iP_{d- d_i}$. Take a basis of this quotient space. Multiplying each of these by $f_1$ gives an additional equation to mod out in $P_{d+d_1}$. He (implicitly, and without proof) claims that these conditions are independent of those from the preceding step.
- The dimension of the quotient of $P_{d+d_1}$ after modding out by all these relations is $$\dim(P_{d+d_1}/\sum_{i=2}^n f_iP_{d + d_1 – d_i}) – \dim(P_d/\sum_{i=2}^n f_iP_{d- d_i}) = \prod_{i=1}^n d_i$$
- This is the number of “independent coefficients after substitution”, and thus has to be the degree of the univariate polynomial claimed to exist in Step 2. (I do not quite understand the reasoning. In step 5 he also talks about an additional condition on $P_{d+d_1}$ imposed by fixing the coefficient of a monomial which I do not quite understand as well, e.g. why that can not be accommodated by fixing a coefficient of the resulting “univariate” polynomial claimed to exist in Step 2.)
- Consequently, the number of solutions of generic $f_1, \ldots, f_n$ is $\prod_{i=1}^n d_i$.