Question

Let \(F\) be a field and \(f=\Sigma_{0 \leq i \leq n} f\left(x^{\prime}\right.\) and \(g=\Sigma_{0 \text { sism }} g, x^{\prime}\) in \(F[x, y]\) have total degrees \(n\) and \(m\) respectively, so that each \(f_{i,} g_{i} \in F(y)\) with degy \(f_{i} \leq n-i\), deg \(g_{i} \leq m-i\). Let \(r=\operatorname{res}_{x}(f, g) \in\) \(F[y]\). Show that each of the \((n+m)\) ! summands contributing to \(r\) has degree at most \(n m\), and hence \(\operatorname{deg}_{y} r \leq n m .\)

Short answer

The degree of resultant \(r\) is at most \(nm\).

Step-by-step solution

Understand the Setup

Let's understand our given setup: We have two polynomials \(f\) and \(g\) in \(F[x, y]\) with total degrees \(n\) and \(m\) respectively. Each coefficient, \(f_i\) and \(g_i\), is a rational function in \(F(y)\), with degree limitations \(\text{deg}_y f_i \leq n - i\) and \(\text{deg}_y g_i \leq m - i\). The resultant \(r = \operatorname{res}_x(f, g)\) is considered in \(F[y]\).

Consider Expansion of Resultant

The resultant \(r\) is expressed as a sum involving determinants formed by the coefficients of \(f\) and \(g\). There are \((n + m)!\) such determinants. Each determinant corresponds to a matrix where columns and rows represent the coefficients of \(f\) and \(g\), respectively.

Determine Degree Contribution

Each matrix in the sum has entries that are polynomials of degree up to \(n - i\) from \(f\) and \(m - i\) from \(g\). Therefore, each entry in the determinant matrix contributes up to \(n \times m\) in terms of the degree of \(y\) when considering degrees collectively.

Analyze Degree of Determinant

The degree of a determinant formed by matrix with polynomial entries is the sum of degrees of any chosen elements per multiplication. Thus, since our biggest contribution can be each entry contributing its maximum possible degree, the maximum degree of the resultant \(r\) is constrained by \(n \times m\).

Conclude the Degree of Resultant

Thus, the determinant matrix that contributes the most to \(r\) will have a degree \((n \times m)\) across all its contributing summands, leading to the conclusion that the overall degree of \(r\) is at most \(n \times m\). Hence, we have \(\text{deg}_y r \leq nm\).

Key concepts

Degree of a Polynomial

The degree of a polynomial is one of the fundamental aspects used in algebra. It refers to the highest power of the variable within the polynomial. For instance, in the polynomial \(3x^5 - 2x^3 + x\), the highest exponent of \(x\) is \(5\), thus the degree is \(5\). Understanding the degree of a polynomial is essential because it can tell you about the behavior of the polynomial function, such as its end behavior, and the number of possible roots.

When dealing with multivariable polynomials, like in the case of polynomials of two variables \(f\) and \(g\) from the example, the concept of total degree becomes important. The total degree is the highest sum of the exponents of variables in any single term. For \(f(x) = x^2y + xy^3 \), the term \(xy^3\) dictates the total degree as \(4\) (since \(1+3=4\)).

Understanding these degrees allows us to also effectively grasp the behavior and interactions of polynomials, especially when considering operations such as addition and multiplication, where degrees can inform us about possible outputs.

Determinants in Algebra

Determinants are essential in algebra, especially when they represent the solvability of systems of linear equations. A determinant is a special number calculated from a square matrix, which gives us important properties about the matrix.

In our example, when the resultant \(r\) is expressed as a sum involving determinants, each determinant corresponds to a matrix formed by coefficients of polynomials \(f\) and \(g\). The determinant can provide crucial information about the roots of the polynomials and their intersection points.

The degree of the determinant—which in context represents the resultant of the polynomials—is critical. It combines the degrees of the polynomials in the determinant sum. In calculating a determinant, each element of the matrix contributes up to its degree in a collective manner, making the resultant degree significant in understanding the maximum contributions possible from the polynomial interactions.

Field Theory

Field theory is a branch of mathematics that is focused on the algebraic structure known as a field. A field is a set with two operations, usually denoted as addition and multiplication, along with their associative, commutative, identity, and inverse properties. The sets of rational numbers, real numbers, and complex numbers are common examples where field properties are apparent.

In the realm of polynomials, being over a field like \(F\) in our example, allows us to manipulate the polynomials using field operations without worrying about undefined operations. This becomes crucial when considering rational functions in \(F(y)\), as it assures that operations on these functions remain consistent and valid.

Field theory plays an important role when dealing with resultants and polynomials as it provides the necessary structure to handle operations and ensures a coherent framework to adequately apply methods such as determinants in polynomial expressions. This understanding helps establish efficient solving methods for polynomial equations and resultants.