Preliminary sample solution to exercise sheet 12 is online
Please remember to register for the exam in the LSF/HISPOS system if you want to participate!
Typo in exercise sheet 12 fixed: Talking about intersections, we should use an intersection symbol instead of a union (exercise 12.1.1.c).
Sample solution to exercise sheet 11 is online
Fixed another typo on exercise sheet 11:V(f1, f2, f3) ≠ ∅ should of course read V(f1, f2, f3) = ∅ in exercise 11.3.5, as indicated by the Nullstellensatz.
(Final) sample solution to exercise sheet 6 is online
Fixed typos on exercise sheet 11:\mathbb{R} should be R in the definition of set intersection and set union in 11.1, and V should be \mathcal{V} (the vanishing set operator) in 11.1.4.
Sample solution to exercise sheet 10 is online
Correction to exercise sheet 10: The corollary ap-1 ≡ 1 mod p from Fermat's little theorem obviously only holds for a's which are not a multiple of p.
Sample solution to exercise sheet 9 is online
Sample solution to exercise sheet 8 is online
Typos in exercise sheet 9 fixed
In 9.1, the definition of the homothetic transformation (scaling) was wrong.
Also, the hint in 9.3 should read "What is φI(1)?", not φI-1(1).
Finally, there is an additional hint for 9.4 part 2.
Preliminary sample solution to exercise sheet 6 is online
Exercise sheet 9 is online
Sample solution to exercise sheet 7 is online
Sample solution to exercise sheet 5 is online
Sample solution to exercise sheet 4 is online
Exercise sheet 7 is online
Absolutely ingenious idea: reverse ordering of the lectures in the list, and moving the overview below
=> Less scrolling for everybody! Hooray!
Exercise sheet 6 is online
Update of the LaTeX template for sample solutions: new shortcuts
Typos in sample solutions for assignment sheet 3 fixed:i should be x in line 4.
Furthermore, principal is the correct term instead of principle in our context.
Sample solutions for assignment sheet 3 are online.
Sample solutions for assignment sheets 1 and 2 are (finally) online.
Typo in assignment sheet 4 fixed.
In exercise 4.4, m and n should be switched.
Note on the programming homework.
You are supposed to use built-in functions for "simple" functions like gcd on polynomials for your implementations.
If some functionality is provided by the environment of your choice, feel free to employ it unless you are explicitly told to reinvent the wheel on this very task.
In particular, if a procedure you need was part of a previous exercise sheet, you can safely assume that you do not need to rely on your own code.
For example, for exercise 4.2, use the root finding capabilities of your system, since this is met by exercise 2.3.
On the other hand, determining the multiplicities of the roots is part of exercise 4.1, which you are supposed to implement here.
This in turn means that you should not use a built-in root solver delivering this information yet.
In case you could not solve 4.1 and, thus, feel unable to complete the implementation: use the built-in anyway.
Be aware that it will result in demerits, but still it's better than nothing at all.
(Minor) typos in assignment sheet 4 fixed.
In the last exercise, the (x, y) of course should be pairs of two real values.
Yet another typo in assignment sheet 2 fixed.
In exercise 2.4, hint 1, the equality holds for the absolute value of the sum over the distances between m and the roots, not for the sum over the absolute values.
Typos in assignment sheet 2 fixed.
Of course, the root solver you are to devise in exercise 2.2 has the same restriction as the variant you have seen in the lecture (with the naive box function): the input has to be squarefree.
Also, there has been a typo in the second hint to exercise 2.4 (power k of ν was missing in the precondition).
Exercise sheet 2 is online.
Submit your source code to practical exercises via email to Alexander
(Get my email address via email).
You are free to use a programming language of your choice; however, we strongly recommend Maple. If in doubt, ask first.
Maple is available on all systems of the computer science and mathematics department (in the computer pool rooms). Do not use Mathematica, since we do not have a license for it.
Do not expect Alexander to be an expert in each and every programming language. He's a proficient C++ coder and, besides, knows C, Java, Python and some computer algebra systems.
In particular, since he's your tutor, you can expect him to be (or become) an experienced Maple user. Do not submit code that does not compile / has syntax errors.
If nothing works, rather hand in pseudocode. We'd like to, but we just do not have the manpower to do a detailed code review or fix your bugs.
