CS 487/687, CM 730 Homework 3

Hensel提升和保理（20分）
1.给定
a（x）= x
4
2倍
3
233x
2个
214x + 85

u0（x）= x
2个
3×2和w0（x）= x
2 + x + 3，

Z [x]中的w（x）使得a = uw。
2.给定
b（x）= 48倍
4
22倍
3 + 47倍
2 + 144

u0（x）= x
2 + 4x + 2和w0（x）= x
2 + 4x + 5

Z [x]中的w（x）使得b = uw。

n是线性独立的，并且L = Zg1 + … + Zgn是它们生成的晶格。证明

n有一个向量g∈L使得
|| x − g || 2
2≤
1个
4
·

|| g1 || 2
2 +···+ || gn || 2
2个
</ s> </ s> </ s> </ s> </ s>

n是线性独立的向量，那么我们将证明我们可以

n−1

1.证明存在一个向量h1∈R
n使得对于所有i∈[m]的hh1，bii = 0
2.使用Gram-Schmidt过程构造R的正交基（h1，…，hn）

ñ

|| 2 = || hj || 2。
3.令H为下式给出的n×n矩阵


H
Ť
1个
H
Ť
2个

H
Ť
ñ



</ s> </ s> </ s> </ s> </ s>
c1 c2···cn
</ s> </ s> </ s> </ s> </ s>
= H·
</ s> </ s> </ s> </ s> </ s>
b1 b2···bn
</ s> </ s> </ s> </ s> </ s>

ci的矩阵

|| 2的上一部分，表明对于每个k∈[m]：
|| ck || 2 =α·|| bk || 2
4.表明HT
·H =α
2个
·I.证明晶格α
2个
（Zb1 +···+ Zbm）和α
2个
（Zc1 +···+ Zcm）是

u∈α
2个
（Zc1 +···+ Zcm）⇔
1个
α2
·HT u∈α
2个
（Zb1 +···+ Zbm）

2个
（Zc1 +···+ Zcm）iff 1
α2
·HT u是α中最短的向量
2个
（Zb1 +···+ Zbm）。

n−1

Problem 4
Issues with the division algorithm – CLO 2.3.5 (20 points)
Let f(x, y, z) = x
3 − x
2y − x
2
z + x, f1(x, y, z) = x
2y − z, and f2(x, y, z) = xy − 1.
1. Compute using the graded lexicographic order:
r1 = remainder of f on division by (f1, f2)
r2 = remainder of f on division by (f2, f1)
Your results should be different. Where in the division algorithm did the difference occur?
2. If r = r1 − r2 in the ideal (f1, f2)? If so, find an explicit expression r = Af1 + Bf2. If not, say why not.
3. Compute the remainder of r on division by (f1, f2). Why could you have predicted your answer before
doing the division?
4. Find another polynomial g ∈ (f1, f2) such that the remainder of division by g by (f1, f2) is non-zero.
Hint: (xy + 1) · f2 = x
2y
2 − 1 whereas y · f1 = x
2y
2 − yz.
5. Does the division algorithm give us a solution to the ideal membership problem for (f1, f2)? Explain.
Problem 5
Different monomial orders – CLO 2.4.10 (20 points)
The following orders are called weight orders. Let u = (u1, . . . , un) ∈ R
n such that u1, . . . , un are positive
real numbers which are linearly independent over Q. We say that u is an independent weight vector. Then,
for α, β ∈ N
n, define:
α >u β ⇔ u · α > u · β
This is the weight order determined by u.
1. Use the corollary of Dickson’s lemma from class to prove that >u is a monomial order.
2. Show that u = (1,

2) is an independent weight vector, so that >u is a weight order on N
2
Problem 6
Monomial Ideals (20 points)
Let I1 = (x
α1
, . . . , xαs ) and I2 = (x
β1
, . . . , xβt ) be monomial ideals of C[x1, . . . , xn], where each αi
, βj ∈ N
n.
1. Show that I1 ∩ I2 is generated by the elements LCM(x
αi
, xβj ).
2. When is I1I2 = I1 ∩ I2? Provide proof of your statement.

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