To be polynomially bounded IN SOMETHING I have a problem of defining and understanding what to be "polynomially bounded in something" means. In fact, the problem is in in, because to be bounded in something does not fits in my mind being a non-native English speaker. I can be "bounded in my growth", my ambitions can be "bounded by my budget", but to be "bounded in the encoding length" or to be "bounded in the size of the input" throws a parse error when I read it.
In particular, we can define an encoding length of some rational number $z=\frac{p}{q}$ with $p \in \mathbb{Z}, q \in \mathbb{Z}_{\ge 1}, gcd(p,q)=1$ as $$\langle z \rangle := 1 + \lceil \log_2(\vert p \vert + 1) \rceil + \lceil \log_2(q) \rceil.$$ And expand the definition for matrices $A \in \mathbb{Q}^{m \times n}$: $$\langle A \rangle := mn + \sum_{i=1}^m\sum_{j=1}^n\langle A_{i,j}\rangle.$$
Now, my book defines a wild number $R := \sqrt{n} \cdot 2^{p(\langle A \rangle + \langle b \rangle)}$, $p$ now being a polynomial, $A \in \mathbb{Q}^{m \times n}$, $b \in \mathbb{Q}^m$, and $n$ generally being a dimension of an Euclidean space. Then, the book says, $\log_2(R)$ is polynomially bounded in the encoding lengths of $A,b$.
Somewhere else in the book I can read that the running time of the algorithm is polynomially bounded in the size of the input. Or, x is bounded in y, or, x is unbounded in y. In all of these cases I simply do not understand how to read and understand that.
Could you please help me to provide a model of how can I transform such an English sentence in the sequence of mathematical statements which are meant by the sentence? Could you please write it in the following manner: "to be bounded IN means <wisdom>. So, taking your example, let $t$ be the running time, $p$ some polynomial, $i$ the input, etc. If t is bounded in  $\langle i \rangle$, then it means <wisdom + math>. Take 2, add 3, and you get 5. The same applies to the other example. Etc."
Please do not redirect me to the question that asks what a polynomially bounded function is. I know Landau, I do not like Landau. I have no problem with Landau, but if you want me to use Landau, please explain in what $f$ is bounded in $f=O(n^2)$. Or in what is $g$ bounded in $g=\Theta(h)$.
I have a problem to understand what a polynomially bounded in X function means. Notation issue + English + transferring.
I would be very thankful if you could explain me the language this way so that I could enrich my mathematical slang for further reading.
 A: This usage comes from "a polynomial in $x$" meaning a polynomial expression where $x$ is the variable. (In turn, this is probably short for "a polynomial in terms of $x$".)
Then, "polynomially bounded in $x$" is meant as "bounded by a polynomial in $x$".
I agree that it is not the easiest terminology to parse if you have not seen it before.
A: Saying that the (time or space) complexity of an algorithm $A$ is polynomially bounded in $x$ means:

*

*We can express the complexity of $A$ as a function of $x$, let's call it $f_A(x)$.


*We can find a polynomial in $x$, let's call it $P(x)$, such that $f_A(x) < P(x)$ for all values of $x$.
So $A$ is being bound by the polynomial $P$, i.e. $P$ provides an upper limit for $A$ at every value of $x$.
For example, let's suppose $A$ is an algorithm that scans a list of length $N$ for its largest value. When $N = 1$, the algorithm takes 7 ms. When $N = 2$, the algorithm takes 17 ms. When $N = 3$, it takes 31 ms, and so on. With some analysis, we show that the run time of $A$ can be written as $f_A(N) = 2N^2 + 4N + 1$. This is a polynomial itself, and it's pretty easy to show that we can choose, for example, $P(N) = 2N^2 + 4N +2$ so that $f_A(N) < P(N)$ for all values of $N$, and hence the time complexity of $A$ is polynomially bounded in $N$, which is the length of its input.
