Question

With the above notation, show that \(\operatorname{dim} V(d, n)=\left(\begin{array}{c}d+n-1 \\ n-1\end{array}\right)\), the binomial coefficient.

Short answer

Question: Show that the dimension of the vector space of homogeneous polynomials in n variables of degree d, denoted as \(V(d, n)\), is given by the binomial coefficient \(\binom{d+n-1}{n-1}\).

Step-by-step solution

Understanding the vector space V(d, n)

The vector space \(V(d, n)\) consists of all homogeneous polynomials of degree d, in n variables. A homogeneous polynomial is a polynomial where all monomials have the same total degree. For example, if we have two variables x and y, a homogeneous polynomial of degree 2 could be written as \(ax^2 + bxy + cy^2\).

Finding a basis for V(d, n)

A basis for \(V(d, n)\) consists of a set of monomials that span the vector space and are linearly independent. We can construct a basis by finding all monomials of degree d in n variables. To do this, we need to find all non-negative integer solutions to the equation \(x_1+x_2+\cdots+x_n = d\), where \(x_i\) represents the exponent of the \(i\)-th variable.

Counting the monomials using combinatorics

We can translate the problem of finding non-negative integer solutions into a combinatorial problem of distributing d identical objects (the exponents) into n distinct boxes (the variables). Imagine d balls and n-1 dividers between them. We can represent a distribution of the d balls into n boxes by placing the dividers between the balls. In such a representation, the number of balls between the dividers corresponds to the power of the variable in the respective monomial. For instance, consider the following scenario with 2 variables (x and y) and the degree d = 4: ``` oo❚oo❚oo ``` Here, we have 4 balls and 1 divider. This corresponds to the monomial \(x^2y^2\) since there are 2 balls before the first divider (x has an exponent of 2) and 2 balls after the first divider (y has an exponent of 2). Hence, we have now reduced the problem to counting the number of ways we can arrange d balls and n-1 dividers. The total number of arrangements is given by choosing n-1 positions out of a total of (d+n-1) positions. This can be found using the binomial coefficient formula:

Writing the result

Applying the binomial coefficient formula to the total number of arrangements of the balls and dividers, we find \(\operatorname{dim} V(d, n) = \binom{d+n-1}{n-1}\).

Key concepts

Homogeneous Polynomials

In algebraic geometry, a homogeneous polynomial is one where all the terms (or monomials) have the same degree. Think of degree as the sum of the exponents in a term. For example, in the polynomial \( 3x^3 + 5x^2y + 7y^3 \), each term is of degree 3. Thus, it's homogeneous.
Homogeneous polynomials are crucial because they maintain their "shape" across transformations, making them useful in many mathematical applications.

• Consider two variables, \( x \) and \( y \). A homogeneous polynomial of degree 2 would include terms like \( ax^2 + bxy + cy^2 \).
• These polynomials are uniform because they simplify under scaling, preserving their degree in each variable.

Understanding this concept is fundamental before diving into more complex structures such as vector spaces of homogeneous polynomials.

Vector Space

A vector space is essentially a collection of vectors that can be added together and multiplied by scalars, while still remaining within the collection. When considering the vector space \( V(d, n) \), it consists of all homogeneous polynomials of degree \( d \) in \( n \) variables.
In mathematical terms, this means every polynomial in \( V(d, n) \) can be represented as a linear combination of basis vectors.

• The basis of this space are monomials, i.e., the simplest building blocks, like \( x^d \) or \( xy^{d-1} \).
• The concept of a basis is tied to linear independence, allowing us to express any vector (polynomial) in the vector space as a sum of these basis vectors.
• Identifying a basis helps in deriving properties like the vector space's dimension, revealing how vast or 'big' the space is.

Understanding vector spaces provides a strong foundation for working with polynomial expressions and other algebraic structures.

Binomial Coefficient

The binomial coefficient, denoted as \( \binom{n}{k} \), represents the number of ways to choose \( k \) elements from a set of \( n \) elements. It's a cornerstone in combinatorial mathematics, often read as "n choose k."
Here's how it applies to the vector space \( V(d, n) \):

• The dimension of \( V(d, n) \) is given by the binomial coefficient \( \binom{d+n-1}{n-1} \).
• This formula counts the number of monomials of degree \( d \) across \( n \) variables, reflecting the number of ways to arrange the polynomial terms.

Binomial coefficients also appear in Pascal's triangle and have properties that simplify many calculations, making them useful in both theoretical and applied problems.

Combinatorics

Combinatorics is the branch of mathematics dealing with counting, arrangement, and combination of elements within sets. When applied to polynomials and vector spaces, it's primarily about counting the number of ways certain polynomial forms can exist.
For example, we can use combinatorics to solve how to distribute degrees among multiple variables in a polynomial:

• We convert the problem into a task of placing dividers between degree units, imagining "balls" (units of degree) and "dividers" (the splits between variable terms).
• For \( V(d, n) \) with degree \( d \), imagine \( d \) balls and \( n-1 \) dividers. The combinatorial challenge is arranging these into a sequence, where each configuration represents a valid polynomial form.

By counting these arrangements using combinatorics, we understand the structure of polynomials deeply, enhancing problem-solving techniques for algebraic geometry.