说明

• 默认多项式为一元多项式。多元多项式需注明
• 默认环含有乘法幺元，子环包含幺元，环同态把幺元映射到幺元，模和代数中幺元的作用必须是恒等的

1 多项式理论基础

1.1 多项式环的构造与通用性质

形式定义、系数环上的多项式环、不定元的泛性质、次数与首项系数、一元与多元的统一框架；多项式作为“有限支撑序列”与作为“表达式”的等价刻画。

1.1.1 一元多项式环

系数环 $R$ 不必为域；$R[x]$ 的环结构、嵌入 $R\hookrightarrow R[x]$、次数映射的性质。

D

• 多项式、系数、次数、项（首项、常数项、k 次次项）
• 零多项式、常数多项式、首一多项式、k 次多项式
• 多项式的加、乘
• 形式导数
• 有理函数

1.1.2 多元多项式环与分次结构

$R[x_1,\dots,x_n]$ 的全次数分级与多次数分级；齐次分量的分解；多元多项式作为一元多项式环上的多项式环。

D

• 变量、多元多项式、多元有理函数
• 形式偏导数
• *Laurent 多项式
• *形式幂级数
• *形式 Laurent 级数

1.1.3 泛性质与代入

对任意 $R$-代数 $S$ 与元素 $s\in S$，存在唯一 $R$-代数同态 $R[x]\to S$，$x\mapsto s$；由此得到多项式函数与形式多项式的区别与联系。

D

• 多项式函数

1.2 多项式环的代数

带余除法的形式与算法含义；整除、相伴、单位的关系；最大公因式与 Bézout 恒等式的环论表述。

1.2.1 整除和带余除法

除式首项系数可逆时的唯一余式；$K[x]$（$K$ 为域）是 Euclid 整环；综合除法与 Horner 算法。

1.2.2 最大公因式与互素

Euclid 算法；$K[x]$ 中理想均为主理想；互素的 Bézout 判别；多项式环与整数环的平行结构及其限度。

D

• 整除、同余
• 商、余式
• 最大公因式（gcd）、最小公倍式（lcm）
• 不可约、相伴

T

• 带余除法、（环）首一带余除法
• Bézout 定理、（环）Bézout 定理
• Euclid 引理、唯一分解定理

1.3 多项式函数与根

D

• 根、重根
• （复）单位根

T

• 因式定理、余式定理
• 根的个数不超过多项式次数
• （无限域）在无限个点恒等时两个多项式恒等。特别地，多项式函数唯一确定多项式
• *域的有限乘法子群是循环的
• （复）代数基本定理及推论

2 多项式的复根 (V.V.P)

• 无极点多项式

• Cauchy 定理

• Laguerre 定理

• 李-杨 圆周定理

...

3 不可约与分解 (V.V.P)

• Eisenstan 判别
• Newton 多边形与 Dumas 判别
• Hilbert 判别
• Cohn 判别
• 反转多项式技术、三项式、四项式判别

...

4 数值方法与性质

• 结式、判别式

• 反转多项式

• 差分

• Lagrange 插值公式

• Viète 定理

• Newton 公式

...

5 特殊多项式 (V.V.P)

• 多元对称多项式
• 整值多项式
• Чебышёв 多项式
• 分圆多项式
• Bernoulli 多项式
• Laguerre 多项式

...

6 多项式的数论

6.1 \Z[x] 上多项式性质

6.2 \F_q[X] 上多项式性质

6.3 代数数理论

• 代数数、代数整数

...

7 交换代数与代数几何初步

...

人名对照

• Bernoulli 伯努利
• Bézout 贝祖、裴蜀
• Cohn 科恩
• Dumas 杜马
• Eisenstein 爱森斯坦、艾森斯坦
• Euclid 欧几里得
• Hilbert 希尔伯特
• Lagrange 拉格朗日
• Laguerre 拉盖尔
• Laurent 洛朗
• Newton 牛顿
• Чебышёв 切比雪夫
• Viète 韦达

...

V.V.P Polynomials Contents

1 Roots of Polynomials 1
1.1 Inequalities for roots 1
1.1.1 The Fundamental Theorem of Algebra 1
1.1.2 Cauchy's theorem 2
1.1.3 Laguerre's theorem 5
1.1.4 Apolar polynomials 7
1.1.5 The Routh-Hurwitz problem 11
1.2 The roots of a given polynomial and of its derivative 12
1.2.1 The Gauss-Lucas theorem 12
1.2.2 The roots of the derivative and the focal points of an ellipse 14
1.2.3 Localization of the roots of the derivative 15
1.2.4 The Sendov-Ilieff conjecture 18
1.2.5 Polynomials whose roots coincide with the roots of their derivatives 20
1.3 The resultant and the discriminant 20
1.3.1 The resultant 20
1.3.2 The discriminant 23
1.3.3 Computing certain resultants and discriminants 25
1.4 Separation of roots 27
1.4.1 The Fourier-Budan theorem 27
1.4.2 Sturm's Theorem 30
1.4.3 Sylvester's theorem 31
1.4.4 Separation of complex roots 35
1.5 Lagrange's series and estimates of the roots of a given polynomial 37
1.5.1 The Lagrange-Biirmann series 37
1.5.2 Lagrange's series and estimation of roots 40
1.6 Problems to Chapter 1 41
1.7 Solutions of selected problems 42
2 Irreducible Polynomials 47
2.1 Main properties of irreducible polynomials 47
2.1.1 Factorization of polynomials into irreducible factors.... 47
2.1.2 Eisenstein's criterion 50
2.1.3 Irreducibility modulo p 51
2.2 Irreducibility criteria 52
2.2.1 Dumas's criterion 52
2.2.2 Polynomials with a dominant coefficient 56
2.2.3 Irreducibility of polynomials attaining small values ... . 58
2.3 Irreducibility of trinomials and fournomials 59
2.3.1 Irreducibility of polynomials of the form $x^n\pm x^m\pm x^p\pm1$ 59
2.3.2 Irreducibility of certain trinomials 63
2.4 Hilbert's irreducibility theorem 65
2.5 Algorithms for factorization into irreducible factors 68
2.5.1 Berlekamp's algorithm 68
2.5.2 Factorization with the help of Hensel's lemma 71
2.6 Problems to Chapter 2 73
2.7 Solutions of selected problems 74
3 Polynomials of a Particular Form 77
3.1 Symmetric polynomials 77
3.1.1 Examples of symmetric polynomials 77
3.1.2 Main theorem on symmetric polynomials 79
3.1.3 Muirhead's inequalities 81
3.1.4 The Schur functions 83
3.2 Integer-valued polynomials 85
3.2.1 A basis in the space of integer-valued polynomials 85
3.2.2 Integer-valued polynomials in several variables 87
3.2.3 The ^-analogue of integer-valued polynomials 88
3.3 The cyclotomic polynomials 89
3.3.1 Main properties of the cyclotomic polynomials 89
3.3.2 The Mobius inversion formula 90
3.3.3 Irreducibility of cyclotomic polynomials 91
3.3.4 The expression for $\Phi_{mn}$ in terms of $\Phi_n$ 93
3.3.5 The discriminant of a cyclotomic polynomial 94
3.3.6 The resultant of a pair of cyclotomic polynomials 95
3.3.7 Coefficients of the cyclotomic polynomials 96
3.3.8 Wedderburn's theorem 97
3.3.9 Polynomials irreducible modulo p 99
3.4 Chebyshev polynomials 100
3.4.1 Definition and main properties of Chebyshev polynomials 100
3.4.2 Orthogonal polynomials 105
3.4.3 Inequalities for Chebyshev polynomials 107
3.4.4 Generating functions . 109
3.5 Bernoulli polynomials 112
3.5.1 Definition of Bernoulli polynomials 112
3.5.2 Theorems of complement, addition of arguments and multiplication 115
3.5.3 Euler's formula 116
3.5.4 The Faulhaber-Jacobi theorem 117
3.5.5 Arithmetic properties of Bernoulli numbers and Bernoulli polynomials 120
3.6 Problems to Chapter 3 125
3.6.1 Symmetric polynomials 125
3.6.2 Integer-valued polynomials 126
3.6.3 Chebyshev polynomials 126
3.7 Solution of selected problems 127
4 Certain Properties of Polynomials 133
4.1 Polynomials with prescribed values 133
4.1.1 Lagrange's interpolation polynomial 133
4.1.2 Hermite's interpolation polynomial 136
4.1.3 The polynomial with prescribed values at the zeros of its derivative 137
4.2 The height of a polynomial and other norms 139
4.2.1 Gauss's lemma 139
4.2.2 Polynomials in one variable 141
4.2.3 The maximum of the absolute value and S. Bernstein's inequality 145
4.2.4 Polynomials in several variables 148
4.2.5 An inequality for a pair of relatively prime polynomials 151
4.2.6 Mignotte's inequality 152
4.3 Equations for polynomials 154
4.3.1 Diophantine equations for polynomials 154
4.3.2 Functional equations for polynomials 162
4.4 Transformations of polynomials 166
4.4.1 Tchirnhaus's transformation 166
4.4.2 5th degree equation in Bring's form 168
4.4.3 Representation of polynomials as sums of powers of linear functions 169
4.5 Algebraic numbers 173
4.5.1 Definition and main properties of algebraic numbers ... 173
4.5.2 Kronecker's theorem 175
4.5.3 Liouville's theorem 176
4.6 Problems to Chapter 4 179
5 Galois Theory 181
5.1 Lagrange's theorem and the Galois resolvent 181
5.1.1 Lagrange's theorem 181
5.1.2 The Galois resolvent 185
5.1.3 Theorem on a primitive element 189
5.2 Basic Galois theory 191
5.2.1 The Galois correspondence 191
5.2.2 A polynomial with the Galois group $S_5$ 195
5.2.3 Simple radical extensions 196
5.2.4 The cyclic extensions 197
5.3 How to solve equations by radicals 199
5.3.1 Solvable groups 199
5.3.2 Equations with solvable Galois group 200
5.3.3 Equations solvable by radicals 201
5.3.4 Abelian equations 204
5.3.5 The Abel-Galois criterion for solvability of equations
of prime degree 208
5.4 Calculation of the Galois groups 212
5.4.1 The discriminant and the Galois group 212
5.4.2 Resolvent polynomials 213
5.4.3 The Galois group modulo p 216
6 Ideals in Polynomial Rings 219
6.1 Hilbert's basis theorem and Hilbert's theorem on zeros 219
6.1.1 Hilbert's basis theorem 219
6.1.2 Hilbert's theorem on zeros 221
6.1.3 Hilbert's polynomial 224
6.1.4 The homogeneous Hilbert's Nullstellensatz for p-fields 231
6.2 Grobner bases 233
6.2.1 Polynomials in one variable 234
6.2.2 Division of polynomials in several variables 235
6.2.3 Definition of Grobner bases 235
6.2.4 Buchberger's algorithm 237
6.2.5 A reduced Grobner basis 239
7 Hilbert's Seventeenth Problem 243
7.1 The sums of squares: introduction 243
7.1.1 Several examples 243
7.1.2 Artin-Cassels-Pfister theorem 248
7.1.3 The inequality between the arithmetic and geometric means 251
7.1.4 Hilbert's theorem on non-negative polynomials $p_4(x, y)$ 253
7.2 Artin's theory 259
7.2.1 Real fields 259
7.2.2 Sylvester's theorem for real closed fields 263
7.2.3 Hilbert's seventeenth problem 266
7.3 Pfister's theory 270
7.3.1 The multiplicative quadratic forms 270
7.3.2 Q-fields 273
7.3.3 Pfister's theorem on the sums of squares of rational functions 274
8 Appendix 279
8.1 The Lenstra-Lenstra-Lovasz algorithm 279
8.1.1 The general description of the algorithm 279
8.1.2 A reduced basis of the lattice 280
8.1.3 The lattices and factorization of polynomials 283
References 289
Index 297