Question

Let \(k\) be a field, \(F \in k\left[X_{1}, \ldots, X_{n}\right], a_{1}, \ldots, a_{n} \in k\). (a) Show that $$ F=\sum \lambda_{(i)}\left(X_{1}-a_{1}\right)^{i_{1}} \ldots\left(X_{n}-a_{n}\right)^{i_{n}}, \quad \lambda_{(i)} \in k $$ (b) If \(F\left(a_{1}, \ldots, a_{n}\right)=0\), show that \(F=\sum_{i=1}^{n}\left(X_{i}-a_{i}\right) G_{i}\) for some (not unique) \(G_{i}\) in \(k\left[X_{1}, \ldots, X_{n}\right]\)

Short answer

Question: Prove that for any polynomial \(F \in k\left[X_{1}, \ldots, X_{n}\right]\) and any elements \(a_1, \ldots, a_n \in k\), the polynomial F can be expressed as a sum of products of the form \(\lambda_{(i)}(X_1-a_1)^{i_1}\ldots(X_n-a_n)^{i_n}\), where \(\lambda_{(i)} \in k\). Moreover, if \(F(a_1, \ldots, a_n) = 0\), then F can also be rewritten as a sum of products of the form \((X_i-a_i)G_i\), where \(G_i \in k\left[X_{1}, \ldots, X_{n}\right]\).

Step-by-step solution

- Observations on F

Note that the polynomial F can be written as the sum of monomials, each of which can be written as a product of the variables \(X_1,\dots, X_n\) raised to some powers, multiplied by a scalar from the field \(k\). Now we will show that each of these monomials can be rewritten in the desired form from (a).

- Rewriting monomials in F

Consider a monomial \(c X_1^{i_1}\ldots X_n^{i_n}\), where \(c \in k\) and \(i_1,\ldots, i_n\) are non-negative integers. Related to Statement (a), let's rewrite this monomial as \(c (X_1-a_1 + a_1)^{i_1}\ldots(X_n-a_n+a_n)^{i_n}\). Now we can expand each term \((X_j - a_j + a_j)^{i_j}\) using the binomial theorem: $$ (X_j-a_j + a_j)^{i_j}= \sum_{k=0}^{i_j} {i_j \choose k} (X_j-a_j)^k a_j^{i_j-k}. $$ By expanding all the terms in the monomial \(cX_1^{i_1}\ldots X_n^{i_n}\) in this way, we obtain a sum of terms of the form \(\lambda_{(i)}(X_1-a_1)^{i_1}\ldots(X_n-a_n)^{i_n}\), where \(\lambda_{(i)} \in k\). Since this is true for each monomial in F, we have shown Statement (a) as the sum of each monomial expansion will result in the desired form.

- Expressing F as a sum of products of the form \((X_i-a_i)G_i\)

For Statement (b), we assume \(F(a_1, \ldots, a_n) = 0\). We will express F as a sum of products of the form \((X_i-a_i)G_i\) by following these steps: 1. Divide \(F\) by \((X_1 - a_1)\). Write \(F\) as \((X_1 - a_1)Q_1 + R_1\), where the remainder \(R_1\) has no terms in \(X_1\). 2. Since \(F(a_1, \ldots, a_n) = 0\), it follows that \(R_1(a_1, \ldots, a_n) = 0\). The first term of the sum will be \((X_1-a_1)Q_1\). 3. For each remaining variable \(X_i\), follow a similar process: a. Divide \(R_{i-1}\) by \((X_i - a_i)\). Write \(R_{i-1}\) as \((X_i - a_i)Q_i + R_i\), where \(R_i\) has no terms in \(X_i\). b. Add the term \((X_i - a_i)Q_i\) to the previous sum from Step 2 (a.), and continue with the next variable, until no variables are left. By the end of this process, we will have expressed \(F\) as a sum of products of the form \((X_i-a_i)G_i\), with \(G_i \in k\left[X_{1}, \ldots, X_{n}\right]\). This shows Statement (b).

Key concepts

Polynomial Expressions

Polynomial expressions are a fundamental part of algebraic geometry. They consist of variables and coefficients, such as those seen in the polynomial \(F\). Each variable can be raised to an integer power, and terms are combined using addition or subtraction. These expressions form the backbone of polynomial functions, allowing us to explore mathematical structures and relationships.

In general, a polynomial expression takes the form \[P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\] where each \(a_i\) is a coefficient from a specific field, like \(k\) in our original exercise. The degree of the polynomial is the highest power of the variable, denoted by \(n\) in this expression.

Understanding how to manipulate these expressions is crucial for unpacking more complex algebraic structures. This involves operations like addition, subtraction, multiplication, and sometimes division, although division can yield expressions that are not polynomials anymore.

Field Theory

In algebraic geometry, field theory underpins our understanding of solutions to polynomial equations. A field is a set equipped with two operations, addition and multiplication, that follow specific rules allowing for division as long as the divisor isn't zero.

The field \(k\) in our problem is a set from which the coefficients of polynomial \(F\) and values \(a_1, \ldots, a_n\) are drawn. It could be rational numbers, real numbers, complex numbers, or a finite field, depending on the context.

Working within a field means that the polynomial expressions maintain properties essential for solving equations or proving theorems. For instance, any non-zero element from the field has a multiplicative inverse, a feature we often need while expressing expressions or in polynomial division. This makes fields incredibly important in higher mathematics as they provide the "playground" for our algebraic conjectures and proofs.

Binomial Theorem

The binomial theorem is a crucial tool that helps us expand expressions raised to a power, such as monomials. In algebraic geometry, this theorem simplifies the process of expanding polynomial expressions.

The theorem states that for any integer \(n\),
\[(x + y)^n = \sum_{k=0}^{n} {n \choose k} x^{n-k} y^k\]

Each expanded term consists of a binomial coefficient \({n \choose k}\), which shows how coefficients in the expanded form are distributed. This makes expansion straightforward and systematic. For example, it was used in our exercise to express powers like \((X_j-a_j+a_j)^{i_j}\) into forms involving lower-order terms, which can be managed more easily.

Mastering this theorem is essential, as it not only helps in expanding expressions but also plays a role in more abstract algebraic forms like rings and fields.

Monomial Expansion

Monomial expansion is the process of breaking down a monomial into its constituent parts. A monomial generally takes the form\(c X_1^{i_1} \ldots X_n^{i_n}\), where \(c\) is a constant and \(i_1, \ldots, i_n\) are powers of the variables.

To expand these, we can apply principles of the binomial theorem, especially when translating higher-degree monomials into sums of simpler terms. For example, rewriting \((X_1-a_1+a_1)^{i_1}\) allows us to break the expression into parts that reveal the influence of each variable or constant involved.

Monomial expansion is extremely helpful for expressing polynomial functions in a different form, which can simplify solving equations or proving identities. The ability to manipulate and expand these expressions is a crucial skill in fields like algebraic geometry, allowing us to handle complex relationships between variables.