How to find the smallest degree polynomial from a family of polynomials?

Question

If I take a family of polynomials, like this one :

$$\mathcal B = ( t+t^2, t^3, t^2 + t^4) $$

I'm searching for proving that the smallest degree polynomial generates the rest of polynomials (as in the demonstration of $K$ field iff $K[t]$ is a principal domain).

So firstly in order to know if this statement is true, I was trying to write $t^3$ as a linear combination of the $2$ other polynomials. But when I wrote the system of equalities, I did not know what degree of $\lambda$ and $\mu$ I should use

( here : $$ t^3 = \lambda(t) (t+t^2) + \mu(t) (t^2+t^4)$$ )

So my question is the following.

Is there any rule to find such $\lambda$ and $\mu$ and what degree for them should I input ?

In result, I'll have at least 4 equations, and if I take for example $\deg( \lambda ) = 3; \deg( \mu ) = 2 $, I have $7$ variables for $6$ equations.

That's annoying...

Maybe I have not understand a fundamental thing. Please feel free to explain me anything that could be helpful.

Answer 1

The equation implies $$t^2=\lambda(t)(1+t)+\mu(t)(t^3+t)$$

Now we have $t^3+t=(t^2-t-2)(t+1)-2$, or $-2=(t^3+t)-(t^2-t+2)(t+1)$, so $$t^2=-\frac{t^2}2(t^3+t)+\left(\frac{t^4}2-\frac{t^3}2+t^2\right)(t+1)$$

For the general method see this.

Answer 2

If we have an ideal of polynomials generated by two polynomials, ie, $I = \langle p_1(x), p_2(x) \rangle$, note that this ideal is equal to $I = I' = \langle gcd(p_1(x), p_2(x)) \rangle$.

This is because $K[x]$ is a principal ideal domain, and the definition of a PID is to be able to express an ideal of multiple generators as that of one generator. Luckily for us $K[x]$ is also a euclidian domain, so we have Euclid's algorithm to compute the generator quite easily.

So in this case, compute $p(x) = gcd(gcd(t + t^3, t^3), t^2 + t^4))$.

Answer 3

Well, the case of polynomials is quite simple as already answered above: The ring $K[t]$ is a principle ideal domain and a generator of the ideal in question can be computed by the Euclidean algorithm.

For practical purposes, you could compute the Gröbner or standard basis of the given collection of polynomials (the ideal they generate). Here is the Singular code:

\$ ring r = 0, (t), lp;

\$ ideal i = t+t2, t3, t2+t4;

\$ std(i);

_[1] = t