Due October 10.

1. Exercise 2.10
   • Part (i), As arithmetic operation in R count addition, negation, subtraction, multiplication. You do not need to count other operations we have discussed such as a nullary function returning zero or copying a ring element from one place in memory to another. Nor do you count arithmetic operations on indices i.
   • Skip part (ii) but redo part (i) showing the sparse polynomial multiplication can be done in nm + n + m + 1 arithmetic operations in R when the function axpy is included. Axpy(a, x, y) updates y by addition of a*x, ( y <-- ax + y ). For some rings of interest, axpy can be done considerably faster than by using a multiply and then an addition.
   • Do part (iii).

2. Exercise 3.3 has 5 parts, 3.5 has 6 parts, 3.9 has two parts. Do at least 6 of the 13 parts. For the biggest challenge do the 3.5 parts.

3. Choose two of the following 4 choices from chapter 4. Two of the choices are to do computations by hand to increase familiarity with modular arithmetic and gcd in integers and polynomial rings. Two of the choices are more mathematically challenging. Do the latter two if you can. Do one or both of the former if you are less familiar with the topic or run into a brick wall with one of the latter two. From the point of view of grading all are equally valid choices.
   • Ch 4, Exercises 6,7,8,9.
   • Ch 4, Exercises 10, 11, 22.
   • Ch 4, Exercise 24.
   • Ch 4, Exercise 32.

5. Ch 5, Exercise 1. Remark: induct on r. Remark: Note that this exercise validates some of the absurd systems in common use such as our scheme for telling time (seconds, minutes, hours, days, years, centuries, ...) and the various non-metric weights and measures used in certain backwaters of the world (ounces, pounds, tons).