Homework #1
Algorithmic Algebraic Geometry
Due Friday, June 3, 2016
Let us set the following notation:
• N, Z, Q, R, R+ , Rn+ , and C respectively denote the positive integers, the integers, the
rationals, the real numbers, the positive real numbers, the positive orthant (or positive
quadrant when n = 2), and the complex numbers.
• We’ll frequently use iff or ⇐⇒ as shorthand for “if and only if”.
• For any ring R, we let R∗ := R \ {0}. (We use “\” for set-theoretic difference.)
• Given any x1 , . . . , xn ∈ C we set x := (x1 , . . . , xn ) and, when x1 · · · xn 6= 0, we also define
Log|x| := (log |x1 |, . . . , log |xn |).
• We let Rn×n denote the collection of all n × n matrices with entries in R.
• For any M ∈ Rn×n with (i, j)-entry mi,j entry, and any vector of variables y = (y1 , . . . , yn ), we define
m
m
m
m
the formal expression y M := (y1 1,1 · · · yn n,1 , . . . , y1 1,n · · · yn n,n )
• A set S ⊆ Rn is path connected iff, for any points x, y ∈ S, there is a continuous function
φ : [0, 1] → S with φ(0) = x and φ(1) = y.
• The hand-outs mentioned below can be found by appending the corresponding file-name
to www.math.tamu.edu/~rojas/
Problems
0: Please read the posted pdfs on: (i) complexity theory notation
(bachshallitcxitynotation4reu.pdf), (ii) the Extended Euclidean Algorithm
(bachshallitextendedeuclideanalgor4reu.pdf and euc4reu.pdf), (iii) the first 4 pages of Viro’s
“Dequantization of Real Algebraic Geometry on Logarithmic Paper” (virologpaper4reu.pdf),
and (iv) the first 2 pages of my notes on estimating root norms (pre.pdf). The other readings
I mentioned earlier are optional.
Note: You need not do Problem 0 first: it may be more helpful do it in parallel with the
others...
1: Please become familiar enough with Linux to be able to open terminal windows and
use commands like ls, vi (or emacs), cd, ssh, scp, and grep. In particular, if you have a
laptop that runs Linux, please install Sage, or at least be comfortable with using ssh to call
Matlab and Maple in a remote session. If you’re completely Linux-averse, then at least find
a way to edit and compile LaTeX files on the operating system you use (and get comfortable
with putty).
Hint: Dr. Phillipson wrote a nice hand-out that will help you with this problem (kai.pdf).
Comment: While SageMathCloud is a nice alternative to installing Sage if you’re not on
Linux, there are some occasional troubles with leaving your files in the cloud instead of
having them on your own device. (I have had students lose access to their work on the cloud
at bad times.) So be forewarned...
2: Please show how to compute 2 using 13 or fewer integer multiplications.
3: (a) Please find integers a and b such that 17a + 437b = 1. Hint: Use the Extended
343
Euclidean Algorithm, as described in Bach & Shallit.
(b) Suppose you know that x17 = α and x437 = β, for some nonzero complex numbers α
and β. Please express x explicitly as a monomial in α and β, without solving for x.
Hint: Find a way to apply your solution to Part (a).
Instructor: J. Maurice Rojas
4:
(a) Please prove that for any A ∈ Zn×n we have log xA = (log |x|)A for any
x = (x1 , . . . , xn ) ∈ (C∗ )n .
(b) Please prove that for any A ∈ Rn×n we have f (x) := xA is invertible (as a function
from Rn+ to Rn+ ) ⇐⇒ det A 6= 0. Hint: When det A 6= 0 there is a very obvious
choice for the inverse of f . The harder direction is then f invertible =⇒ det A 6= 0.
A good approach is to prove the contrapositive: assume det A = 0 and then prove
that f must fail to be injective (which implies a failure of invertibility). In
particular, when det A = 0, you can use a nonzero vector in the left-null space of A
to generate numerous distinct x with xA = (1, . . . , 1).
We usually call a substitution of the form x := y A a monomial change of variables.
5: Recall the following formula due to Euler, first
√ published in 1748:
e
√
−1θ
= cos(θ) +
−1 sin(θ),
where θ ∈ R.
√
Let Hu,v denote the number of complex roots√of e 2y = 1 in the horizontal strip R × [u, v].
≤ C for all u < v. Hint: First find an
Please find a positive integer C with √Hu,v − 2 · v−u
2π
2y
explicit formula for all the roots of e = 1.
6:
Please prove the claims made in Example 1.2 of the handout www.math.tamu.edu/~rojas/pre.pdf,
i.e., derive (and prove) an explicit description of the amoeba of the n-variate binomial
a
a
a
a
c1 (x11,1 · · · xnn,1 ) + c2 (x11,2 · · · xnn,2 ).
7: Let p ∈ N be any prime and define F
p — the finite field with p elements — to be Z/pZ,
i.e., the integers mod p.
(a) Please find all the square roots of 2 in F17 .
(b) Please prove that a(p−1)/2 6= 1 (mod p) implies that a has no square root in Fp (unless a
is divisible by p).
(c) Please find a function η(d, p) such that such that xd = c has a root in Fp ⇐⇒ cη(d,p) = 1.
Hint: It may help for you to be aware of 3 facts:
• Fermat’s Little Theorem + a Finite Field Fact: F∗p is a cyclic group of order p − 1. For
instance, for p = 97, any integer in {1, . . . , 96} can be obtained as a power of 5 mod
97.
• Lagrange’s Theorem: H a subgroup of a finite group G =⇒ #H|#G. For instance, the
∗
powers of 64 mod 97 form a group with 8 elements (a subgroup of F97
), and 8 indeed
divides 96.
• Bézout’s Lemma: For any relatively prime integers u and v, there are integers a and b
with au + bv = 1.
8:
Suppose now that d ∈ Z and c ∈ R.
(a) Please find precise conditions on (c, d) under which xd = c has a positive root.
(b) Please find precise conditions on (c, d) under which xd = c has a real root.
9: Please prove that, given any finite subset X ⊂ R , the set R
2
2
\ X is path-connected. In
particular, please outline your own method to build an explicit path in R2 \ X connecting
any two given points p and q. You should think of your construction as an algorithm you
could actually implement. Note: I’m really just looking for a clear, concise explanation of
a method of your own that works. (I’ve seen excellent solutions within 1 page.)
NOTE: Please feel free to e-mail comments, questions, and/or corrections.
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