Question

Let \(F\) be a field, \(f, g \in F[x, y]\) nonzero with \(\operatorname{deg}_{x} f, \operatorname{deg}_{x} g \leq n, \operatorname{deg}_{y} f, \operatorname{deg}_{y} g \leq d\), and \(\operatorname{lc}_{x}(f)=\) \(\mathrm{lc}_{x}(g)=1\). Suppose that \(\operatorname{gcd}(f(x, u), g(x, u)) \neq 1\) for at least \(2 n d+1\) values \(u \in F\). Conclude that \(\operatorname{deg}_{x} \operatorname{gcd}(f, g)>0\).

Short answer

The x-degree of gcd(f, g) is greater than zero.

Step-by-step solution

Understanding the Setting

We start by understanding that we have two polynomials \( f \) and \( g \) over a field \( F \) in two variables, with each polynomial having a degree in \( x \) and \( y \) at most \( n \) and \( d \) respectively. The condition \( \operatorname{lc}_{x}(f) = \operatorname{lc}_{x}(g) = 1 \) implies that both polynomials are monic in \( x \).

Specialization at \( u \)

When substituting \( y \) by a specific value \( u \) from \( F \), we obtain one-variable polynomials \( f(x, u) \) and \( g(x, u) \). The problem states that \( \operatorname{gcd}(f(x, u), g(x, u)) eq 1 \) for at least \( 2nd+1 \) values of \( u \).

Applying the Degree Bound

Using the degree bounds \( \operatorname{deg}_{x} f \leq n \) and \( \operatorname{deg}_{x} g \leq n \), and taking into account that the greatest common divisor condition fails for more than the number of coefficients in a polynomial of degree \( nd \), we set the scene for applying properties of resultants or polynomial degree theory.

Utilizing the Resultant

The resultant \( R(f, g) \) of two polynomials is non-zero if their \( x \)-variable resultant has all roots in common. Given \( \operatorname{gcd}(f(x, u), g(x, u)) eq 1 \) for many \( u \), it implies that the nontrivial \( x \)-degrees must be common for these many \( u \).

Concluding on Degree of GCD

If the resultant \( R(f, g)(u) \) is zero for more than \( nd \) values of \( u \), then by the polynomial identity principle, \( R(f, g) \) itself is zero. Therefore, the \( x \)-degree of the \( \operatorname{gcd}(f, g) \) must be more than zero, confirming the conclusion of positive degree common factor.

Key concepts

Resultant

The concept of the resultant plays a crucial role in understanding the relationship between two polynomials across one variable. When dealing with two polynomials, say \(f(x)\) and \(g(x)\) over a field \(F\), we want to know if they have a common root or more generally, common factors.

• The resultant \(R(f, g)\) is a determinant expression that is computed from the coefficients of \(f(x)\) and \(g(x)\).
• If \(f\) and \(g\) do not share a common factor, the resultant \(R(f, g)\) is non-zero.
• Conversely, if the resultant is zero, there is a nontrivial common factor, meaning they share a root.

When a polynomial \(R(f, g)(u)\) is zero for enough values of \(u\), it implies common roots, leading us to infer the presence of a non-zero polynomial you can visualize, like a higher-degree greatest common divisor (GCD) than expected. This knowledge underpins the conclusion that if \(f(x, u)\) and \(g(x, u)\) have common factors for more than a calculated number of substitutions \(u\), then \(R(f, g)\) must indeed be zero, indicating a higher degree of commonality.

Degree Bound

The degree bound in polynomial equations is a concept that assists in understanding how the degree of a polynomial constrains the behavior of its solutions. In this exercise, it's crucial to notice how the bounds on the degree in \(x\) and \(y\) apply:

• Both \(f\) and \(g\) have degrees in \(x\) less than or equal to \(n\), and degrees in \(y\) less than or equal to \(d\).
• When examining disturbances in solutions \(u\), we look for situations where the common factor between \(f(x,u)\) and \(g(x,u)\) is visible more frequently than allowed.

This frequency, set at \(2nd + 1\), exceeds the potential degree for a sole-variable polynomial built from \(f\) and \(g\). Such excessive presence mandates a re-evaluation of assumed degree bounds, hinting that the effective degree must be greater, establishing a nontrivial common factor.

Polynomial Identity Principle

The Polynomial Identity Principle offers a profound understanding of why a polynomial with many zero values must itself be zero. This principle is particularly instrumental when multiple substitutions result in a zero outcome

• If a polynomial in one variable is zero for more values than its degree, the entire polynomial is identically zero.
• This principle helps prove that common factors exist if \( f(x, u) \) and \( g(x, u) \) produce \( \operatorname{gcd}(f(x, u), g(x, u))\) repeatedly without reaching quick exhaustion of possibilities.

In simple terms, the principle supports the deduction of a positive degree common polynomial factor if repeating the zeros occurs over \(2nd + 1\) times. This conclusion arises because it's impossible for another separate factor to absorb this repetition without resulting in a genuine zero polynomial degree, thus confirming the alignment in polynomial identities.