Sind f=am t^m+…+a0, g=bn t^n+…+b0 ∈K[t] Polynome mit deg f=m, deg g=n, so ist die Resultante von

step 1 Hello there. Okay, so for this exercise we got two linear maps, F and G, defined as follows. We

Sind f=am t^m+…+a0, g=bn t^n+…+b0 ∈K[t] Polynome mit deg...

Sind $f=a_{m} t^{m}+\ldots+a_{0}, g=b_{n} t^{n}+\ldots+b_{0} \in K[t]$ Polynome mit $\operatorname{deg} f=m, \operatorname{deg} g=n$, so ist die Resultante von $f$ und $g$ definiert durch $$ \left.\operatorname{Res}_{f, g}:=\operatorname{det}\left(\begin{array}{ccccccc} a_{0} & & \cdots & \cdots & & a_{m} & & \\ & \ddots & & & & & \ddots & \\ & & a_{0} & & \cdots & \cdots & & a_{m} \\ b_{0} & \cdots & \cdots & b_{n} & & & & \\ & & & & b_{0} & \cdots & \cdots & b_{n} \end{array}\right)\right\} n \text { Zeilen } $$ Zeigen Sie die Äquivalenz der folgenden Aussagen: i) $\operatorname{Res}_{f, g}=0$ ii) $f, t f, \ldots, t^{n-1} f, g, t g, \ldots, t^{m-1} g$ sind linear abhängig. iii) Es existieren $p, q \in K[t], p, q \neq 0$, mit deg $p \leq n-1, \operatorname{deg} q \leq m-1$ und $p f=q g$ Mit etwas Teilbarkeitstheorie von Polynomen kann man zeigen, dass i) bis iii) äquivalent sind $z u$ iv) $f$ und $g$ haben einen gemeinsamen nichtkonstanten Teiler $h \in K[t]$. Insbesondere ist also $\operatorname{Res}_{f, g}=0$, falls $f$ und $g$ eine gemeinsame Nullstelle haben, und im Fall $K=\mathbb{C}$ gilt: $\operatorname{Res}_{f, g}=0 \Leftrightarrow f$ und $g$ haben eine gemeinsame Nullstelle.

Sind $f=a_{m} t^{m}+\ldots+a_{0}, g=b_{n} t^{n}+\ldots+b_{0} \in K[t]$ Polynome mit $\operatorname{deg} f=m, \operatorname{deg} g=n$, so ist die Resultante von $f$ und $g$ definiert durch
$$
\left.\operatorname{Res}_{f, g}:=\operatorname{det}\left(\begin{array}{ccccccc}
a_{0} & & \cdots & \cdots & & a_{m} & & \\
& \ddots & & & & & \ddots & \\
& & a_{0} & & \cdots & \cdots & & a_{m} \\
b_{0} & \cdots & \cdots & b_{n} & & & & \\
& & & & b_{0} & \cdots & \cdots & b_{n}
\end{array}\right)\right\} n \text { Zeilen }
$$
Zeigen Sie die Äquivalenz der folgenden Aussagen:
i) $\operatorname{Res}_{f, g}=0$
ii) $f, t f, \ldots, t^{n-1} f, g, t g, \ldots, t^{m-1} g$ sind linear abhängig.
iii) Es existieren $p, q \in K[t], p, q \neq 0$, mit deg $p \leq n-1, \operatorname{deg} q \leq m-1$ und $p f=q g$

Mit etwas Teilbarkeitstheorie von Polynomen kann man zeigen, dass i) bis iii) äquivalent sind $z u$
iv) $f$ und $g$ haben einen gemeinsamen nichtkonstanten Teiler $h \in K[t]$.
Insbesondere ist also $\operatorname{Res}_{f, g}=0$, falls $f$ und $g$ eine gemeinsame Nullstelle haben, und im Fall $K=\mathbb{C}$ gilt: $\operatorname{Res}_{f, g}=0 \Leftrightarrow f$ und $g$ haben eine gemeinsame Nullstelle.

Key Concepts

Polynomial Divisibility and Common Factors

Polynomial divisibility concerns whether one polynomial can be 'divided' by another without a remainder, leading to the concept of common factors. In this context, the existence of nonzero polynomials of bounded degree such that one polynomial becomes a multiple of the other indicates that the two given polynomials have a nonconstant common divisor. This idea is central to many areas in algebra, particularly in understanding greatest common divisors in polynomial rings.

Common Zeros and Field Considerations

When working over an algebraically closed field like the complex numbers, the notion of a common factor is equivalent to the existence of a common zero. Thus, the vanishing of the resultant signals that the two polynomials share at least one root. This connection between algebraic properties of polynomials and their geometric interpretation in terms of zeros is fundamental to algebraic geometry and complex analysis.

Sylvester Matrix Construction

The Sylvester matrix is a structured matrix formed by placing the coefficients of the two polynomials in a specific pattern, so that its determinant (the resultant) reveals properties related to their common factors. Its rows consist of shifted sequences of the coefficients of each polynomial, with the number of shifts corresponding to the degree of the alternate polynomial. This construction links linear algebra with polynomial algebra.

Resultant

The resultant is a scalar quantity associated with two polynomials that encapsulates whether they have a nontrivial common factor. It is defined as the determinant of a specific matrix (the Sylvester matrix) constructed from the coefficients of the polynomials. A zero resultant indicates that the polynomials share a common factor or, equivalently, at least one common root (when considered over an algebraically closed field).

Linear Dependence in Polynomial Spaces

Linear dependence in the context of polynomial spaces refers to the situation where a set of polynomials, usually constructed by multiplying a given polynomial by powers of the variable, can be expressed as linear combinations of one another. In this problem, the condition that certain multiples of the two given polynomials are linearly dependent is equivalent to the vanishing of the resultant. This equivalence provides a bridge between the algebraic concept of determinants and the geometric idea of shared roots or factors.

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