Why $\mathcal{O}(n^2)$ multiplication of coefficient required for canonical form of polynomial?

I was going to a textbook and am not able to understand the following:

Let $ F(x)$ is given as a product $ F(x) = \sum_{i=0}^{n} (x – a_i)$ . Transforming $ F(x)$ to its canonical form by consecutively multiplying the $ i$ th monomial with the product of the first $ i-1$ monomials requires $ \mathcal{O}(n^2)$ multiplications of coefficients.

A canonical form of polynomial is $ \sum_{i=0}^{n} c_i x_i$ .

Why $ \mathcal{O}(n^2)$ ?

Degree bounds and coefficient size in elimination theory?

Suppose we have polynomials is of form $ $ h_1(x_1,\dots,x_n)-c_1\in\mathbb Z[x_1,\dots,x_n]$ $ $ $ \vdots$ $ $ $ h_n(x_1,\dots,x_n)-c_n\in\mathbb Z[x_1,\dots,x_n]$ $ where $ h_1(x_1,\dots,x_n),\dots,h_n(x_1,\dots,x_n)\in\mathbb Z[x_1,\dots,x_n]$ are homogeneous polynoials of degree $ d$ with each $ x_i$ degree $ 1$ (that is only monomials of form $ x_{i_1},\dots,x_{i_d}$ on the condition that each coefficient is random in $ [-B,B]$ with $ c_1,\dots,c_n$ being of absolute value at most $ B^{1+\frac d{2^d-1}}$ then if you use elimination theory we can successively eliminate variables and reduce to an univariate polynomial.

Let the absolute value of the maximum coefficient size $ B’$ of final univariate polynomial in reduced form (no common gcd of coefficients) and the let the degree be $ d’$ .

Then can $ B’^{\frac1{d’}+\epsilon}>B^{\frac1{d(2^d-1)}}$ hold at arbitrarily small $ \epsilon>0$ as $ B$ increases and $ d$ is fixed?

O(1) algorithm to get approximate number that is larger as a binomial coefficient

Ideally I need a to calculate the binomial coefficient $ {p \choose n}$ . But since the fastest algorithm to do this is an $ \mathcal{O}(n)$ algorithm I would look to do something different. I don’t really care for the exact number as long as it is reasonably close and is at least equal or larger then the actual number. My number should be within a factor of 2 of actual binomial coefficient. Is there an $ \mathcal{O}(1)$ way to compute such a number?

Correlation coefficient of a WSS process

Could someone please tell me why the correlation coefficient of a WSS process is :

$ $ \rho_{XX}(\tau) = \frac{\operatorname{K}_{XX}(\tau)}{\sigma^2} = \frac{\operatorname{E}[(X_t – \mu)(X_{t+\tau} – \mu)]}{\sigma^2}$ $

Instead of :

$ $ \rho_{XX}(\tau) = \frac{\operatorname{K}_{XX}(\tau)}{\sigma_{t}\sigma_{t+\tau}} = \frac{\operatorname{E}[(X_t – \mu)(X_{t+\tau} – \mu)]}{\sigma_{t}\sigma_{t+\tau}} $ $

?

Where :$ ~~~\sigma_{t}^2=\sigma^2=\operatorname{E}[X^2(t)]-\operatorname{E}^2[X(t)]$ .

Stated in another way , does a WSS process imply that :

$ $ \operatorname{E}[X^2(t+\tau)]=\operatorname{E}[X^2(t)]$ $ ?

(https://en.wikipedia.org/wiki/Autocovariance)

Well posedness of wave equations whith time depending coefficient

Let us consider the folooiwng wave equation \begin{array}{rrr} y_{tt}=y_{xx}+a(t,x)y_{t} & \text{in} & (0,T)\times (0,1), \ y(0,t)=0\text{ , }y(t,1)=0 & \text{in} & (0,T), \ y(0,x)=y_{0}(x)\text{ , }y_{t}(0,x)=y_{1}(x) & \text{in} & (0,1). \end{array} Assume that for example that $ \left( {{y_0},{y_1}} \right) \in H_0^1(0,1) \times {L^2}(0,1)$ (resp. $ \left( {{y_0},{y_1}} \right) \in {L^2}(0,1) \times {H^{ – 1}}(0,1)$ ). In the case of time depending coefficients semigroups theory fails, we have to deal with this problem otherwise. I saw some books and I found that $ a$ must be $ C^1$ in time. Is this the optimal assumption? Because I could solve this problem by characterestics and I didn’t need to this assumption, it was just $ $ a \in {L^2}((0,T) \times (0,1))$ $ . Any suggestions?. Thank you.

Is the following variant of the Universal Coefficient Theorem valid?

A version of the Universal Coefficient Theorem that relates the integer cohomology of a group $ G$ to its cohomology with coefficients in an abelian group $ M$ is as follows:

$ H^n(G,M) = H^n(G,\mathbb Z) \otimes M \quad \times \quad\text{Tor}_1^{\mathbb Z}(H^{n+1}(G,\mathbb Z), M)$

It is assumed in this expression that $ G$ acts trivially on coefficients. Now suppose $ G$ is an extension of the group $ \mathbb Z_2$ by some group $ G_0$ and $ M = \mathbb Z_2$ . I am interested in finding the cohomology of $ G$ with coefficients in the module $ \mathbb Z^{sgn}$ , whose coefficients are twisted by the $ \mathbb Z_2$ factor, while $ G_0$ has trivial action. I wanted to know if the following statement, which looks like the statement of the UCT, is actually valid:

$ H^n(G,\mathbb Z_2) = H^n(G,\mathbb Z^{sgn}) \otimes \mathbb Z_2 \quad \times \quad\text{Tor}_1^{\mathbb Z}(H^{n+1}(G,\mathbb Z^{sgn}), \mathbb Z_2)$

(So just to be clear, the l.h.s. has trivial action on $ \mathbb Z_2$ , while the r.h.s. has nontrivial action on $ \mathbb Z^{sgn}$ . I have checked this by hand for some simple examples and it works out, but I don’t have any proof. )

how to get coefficient list from a polynomial with negative powers

Say a polynomial x^2-x^(-2), I need to extract its coefficient. I tried the command CoefficientList[x^2-x^(-2)], but no result comes out.

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I am confused why it happens. How can I get the desired result {-1,0,0,0,1}?

It should be an easy question, but I do not know the suitable command. Would you please give me some tips?