Algorithms for elementary operations using other elementary operators
Throughout this answer, when we say we will be using some operations, it implies that no other operations is allowed. We also assume 1 is available as an operand. 

The above clarification is considered as an improvement of original statement of problems. There will be further modification/clarification of the original problems, which have to be made because of the ambiguity of the original problems.



Here is the answer to part (i)


  Part (i). The product of two non-zero numbers using only reciprocation operation and subtraction operation. 


Please note I have replaced by addition by subtraction; otherwise, it is impossible. Also I removed the specification of $n$-bits, which seems distracting and harmful; otherwise, $\dfrac13$ is problematic. I have also added the condition that the numbers are non-zero. Otherwise, we have to add an operation that checks whether a number is non-zero and acts accordingly. Also 1 have to be available. Otherwise, we cannot produce 1, which is needed in the final operation.

Let the two non-zero numbers are $A, B$.


we get addition by $X+Y=X-((Y-Y)-Y)$.
we get 4=1+1+1+1.
we get $X\to \dfrac {X^2}4$ by $\dfrac {X^2}4=X+\dfrac1{\dfrac1{X-4}-\dfrac1X}$.  
We get multiplication by $AB=\dfrac{(A+B)^2}4-\dfrac{(A-B)^2}4$.


This answer at mathoverflow uses division by 2 or multiplication by $\dfrac12$ without explanation, which can be considered as a minor flaw. In fact, we can do division by 2 as $\dfrac A2=\dfrac1{\dfrac1A+\dfrac1A}$.




  Part (ii). The square of a polynomial with given coefficients using only reciprocation and subtract operation.


Since the coefficient of the square of a polynomial is the sum of products of various coefficients of that polynomial, this part can be done easily as a consequence of part (i).




  Part (iii). The product of polynomials with given coefficients using only squaring and subtraction operations and division by 2.


We get addition by $X+Y=X-((Y-Y)-Y)$. We get multiplication by $XY=\dfrac{(X+Y)^2-X^2-Y^2}2$. Since the coefficient of the square of a polynomial is the sum of products of various coefficients of that polynomial, we can compute the product of two polynomials.

Note that division by 2 is needed. Otherwise, it can be proven that it is not possible to do multiplication although we have $2XY=(X+Y)^2-X^2-Y^2$.