Notes for Intro to Algebraic Geometry

Ting Gong

Started Mar. 24, 2019. Last revision April 23, 2019 Contents

1 Notes: Geometry, Algebra and Algorithm 1 1.1 Polynomials and Affine Space ...... 1 1.2 Affine Varieties ...... 2 1.3 Parametrization of Affine Varieties ...... 3 1.4 Ideals ...... 3 1.5 Polynomial of One Variable ...... 5

2 Notes: Grobner Bases 7 2.1 Introduction ...... 7 2.2 Orderings on Monomials ...... 7 2.3 A Division Algorithm in k[x1, . . . , xn]...... 8 2.4 Monomial Ideals and Dickson’s Lemma ...... 8 2.5 The Hilbert Basis Theorem and Grobner Bases ...... 9 2.6 Properties of Grobner Bases ...... 11 2.7 Buchberger;s Algorithm ...... 12 2.8 First Applications of Grobner Bases ...... 12

i Chapter 1

Notes: Geometry, Algebra and Algorithm

1.1 Polynomials and Aﬃne Space

We are going to studt polynomials of a ﬁeld in this course, since linear algebra works on any of them. Examples are R, Q, C etc.

α1 αn Deﬁnition 1.1.1. A monomial in x1, . . . , xn is a product of the form x1 . . . xn , where α1, . . . , αn are nonnegative integers. The total degree of the monomial is α1 + ... + αn

α α1 αn In the future, we denotex = x1 . . . xn , α = (α1, . . . , αn), and |α| = α1 + ... + αn.

Definition 1.1.2. A polynomial f in x1, . . . , xn with coefficients in a field F is a finite linear P α combination of monomials. We denote f = α aαx , aα ∈ F . The set of all polynomials in x1, . . . , xn with coefficients in F is denoted as F (x1, . . . , xn) (A field extension of F .

We have examples Q(x), R(x, y), C(x, y, z). P α Definition 1.1.3. Let f = α aαx be a polynomial in F (x1, . . . , xn). Then we call aα α α the coefficient of monomial x . If the coefficient aα 6= 0, we call aαx a term of f. The total degree of f 6= 0 is denoted as deg(f), is the maximum |α| such that aα is nonzero. The total degree of the zero polynomial is undefined.

Then an example.

Remark 1.1.4. The sum and product of two polynomials are polynomials. We say f devides a polynomial g if g = fh for some h ∈ F (x1, . . . , xn) nonzero. We can show F [x1, . . . , xn] to be a polynomial ring.

+ Definition 1.1.5. Given a field F , n ∈ Z , we define the n-dimensional affine space over n F to be F = {(x1, . . . , xn)|x1, . . . , xn ∈ F }.

1 n 1 2 Examples are R , for F we call an affine line, and F we call an affine plane. We can connect a polynomial with an affine space by having the evaluation map f : F n → F .

Proposition 1.1.6. Let F be an infinite field and f ∈ F [x1, . . . , xn]. Then f = 0 in n F [x1, . . . , xn] iff f : F → F is the zero function. (It might not be true in finite field.) Proof. (⇒) Obvious. (⇐) We prove by induction. Base case: n = 1, by Fundamental Theorem of Algebra, f ∈ F [x] of degree m has m roots. Let f(a) = 0, ∀a ∈ F . Since F is infinite, f has infinitely many roots. Thus f = 0. Assume this being true to n − 1. Let f ∈ F [x1, . . . , xn] be a zero function. We can write PN i f = i=1 g1(x1, . . . , xn−1)xn. Where gi ∈ F [x1, . . . , xn−1]. We want to show that gi is a n−1 zero polynomial in F [x1, . . . , xn]. Let (a1, . . . , an−1) ∈ F . We have f(a1, . . . , an−1, xn) ∈ F [xn]. By induction hypothesis, f(a1, . . . , an−1, xn) = 0 in F [xn]. Thus, we see that gi(a1, . . . , an−1) = 0. Therefore, by our inductive assumption, gi is the zero polynomial in F [x1, . . . , xn−1]. Therefore, f is the zero polynomial in F [x1, . . . , xn]. ut

n Corollary 1.1.7. Let F be a infinite field. f, g ∈ F [x1, . . . , xn]. Then f = g iff f : F → F , and g : F n → F are the same function.

Proof. (⇒) Obvious. (⇐) Assume f, g are the same function. f − g is a zero function. Therefore by our last proposition, f − g = 0, thus f = g. ut

Theorem 1.1.8. (Fundamental Theorem of Algebra) Every nonconstant polynomial f ∈ C[x] has a root in C. 1 Proof. Assume there is no such root. Let g = f . Then we want to show that g is bounded and entire, and by Liouville’s Theorem we have f is constant. g is clearly entire since it 1 is the inverse of the polynomial. Consider g = n n . As n → ∞, g → 0. |z| (an+an−1/z+...+a0/z Thus, ∃R > 0, such that ∀|z| > R, |f(z)| < 1. Since |z| ≤ R is compact, there is a maximum M. Thus f(z) is bounded by either M or 1. Thus we have our desired outcome. ut

Deﬁnition 1.1.9. We say that F is algebraically closed if every nonconstant polynomial in F [x] has a root in F .

1.2 Aﬃne Varieties

Definition 1.2.1. Let F be a field, and let f1, . . . , fn be polynomials in F [x1, . . . , xn]. Then n V (f1, . . . , fn) = {(a1, . . . , an) ∈ F |fi(a1, . . . , an) = 0, ∀1 ≤ i ≤ s}. We call V the affine variety defined by f1, . . . , fn. Therefore, the affine variety is the set of all solutions of systems. We see examples such as V (x2 + y2 − 1), V (xy − x3 + 1), V (z − x2 − y2) and several others. Then the twisted

2 cubie. The dimensions of the aﬃne varieties is not intuitive. Consider the linear variety of F n of m equations, by linear algebra, we know the dimension is not necessarily n − m, but n − r, with r being the rank. The key is ”linear Independence”. Consider Lagrange Multiplier Theorem. Moreover, an aﬃne variety can be empty.

Lemma 1.2.2. If V,W are aﬃne varieties, then V ∪ W and V ∩ W are aﬃne varieties.

Proof. Let V = V (f1, . . . , fn), W = V (g1, . . . , gm). For V ∩ W , we see that f1, . . . , fn = 0 and g1, . . . , gm = 0. Thus f1, . . . , fn = g1, . . . , gm = 0. Thus V ∩W = V (f1, . . . , fn, g1, . . . , gm). Therefore, V ∩ W is an aﬃne variety. Consider V ∪ W , we want to show that V ∪ W ⊆ V (figj). If fi = 0, then we know that figj = 0. Similarly, if gj = 0, then figj = 0. Then we check V (figj) ⊆ V ∪ W . If figj = 0, then either fi = 0, where figj ∈ V , or gj = 0, where figj ∈ W . Therefore, V ∪ W = V (figj) thus is also a variety. ut

1.3 Parametrization of Aﬃne Varieties

Definition 1.3.1. Let F be a field, a rational function in t1, . . . , tn with coefficients in F is a quotient f/g of two polynomials f, g ∈ F [t1, . . . , tn], where g 6= 0. The set of all rational functions in t1, . . . , tn with coefficients in F is denoted F (t1, . . . , tn).

0 0 0 0 Notice that two rational functions f/g = f /g iﬀ g f = gf . And F (t1, . . . , tn) forms a ﬁeld.

n Deﬁnition 1.3.2. Let V = V (f1, . . . , fn) ⊆ F be a variety. Then a rational parametric representation of V consists of rational functions r1, . . . , rn ∈ F (t1, . . . , tn) such that the points xn = rn(t1, . . . , tm), ∀n lie in V . If r1, . . . , rn are polynomials, then we call it a polynomial parametric representation of V . The original equations of V are called implicit representation of V .

Deﬁnition 1.3.3. If an aﬃne variety can be parametrized by a rational parametric repre- senation, then they are called unirational.

Notice that given a parametric representation of an affine variety, we can always find the defining equations. We will prove this later in the course. Next, we look at examples, x2 + y2 = 1 and V (y − x2, z − x3).

1.4 Ideals

Deﬁnition 1.4.1. A subset I ⊆ F [x1, . . . , xn] is an ideal if it satisﬁes: (i) 0 ∈ I (ii) If f, g ∈ I, then f + g ∈ I. (iii)If f ∈ I, h ∈ F [x1, . . . , xn], then hf ∈ I.

3 Deﬁnition 1.4.2. Let f1, . . . , fs be polynomials in F [x1, . . . , xn]. Then we set

s X hf1, . . . , fsi = hifi|h1, . . . , hs ∈ F [x1, . . . , xn] i=1

We call it ideal generated by f1, . . . , fs.

Lemma 1.4.3. Let f1, . . . , fs be polynomials in F [x1, . . . , xn], then hf1, . . . , fsi is an ideal of F [x1, . . . , xn]. Ps Proof. First, 0 ∈ hf1, . . . , fsi since 0 = i=1 0fi. Then, Let f, g ∈ hf1, . . . , fsi, f = Ps Ps Ps i=1 hifi, and g = i=1 kifi, f + g = i=1(hi + ki)fi. Thus f + g ∈ hf1, . . . , fsi. Finally, Ps let x ∈ F [x1, . . . , xn], then xf = i=1(xhi)fi. Thus xf ∈ hf1, . . . , fsi. Thus, hf1, . . . , fsi is an ideal of F [x1, . . . , xn]. ut

Definition 1.4.4. An ideal I is finitely generated if ∃f1, . . . , fs ∈ F [x1, . . . , xn] such that I = hf1, . . . , fs}. And we say that f1, . . . , fs are basis of I. Next, we show that a variety depends only on the ideal generated by defining functions.

Proposition 1.4.5. If f1, . . . , fS and g1, . . . , gt are bases of the same ideal in F [x1, . . . , xn], then we have V (f1, . . . , fS) = V (g1, . . . , gt).

Proof. Take ag ∈ V (g1, . . . , gt), f ∈ hf1, . . . , fsi, and sinceg1, . . . , gt and f1, . . . , fS span the Ps Ps same ideal, f ∈ hg1, . . . , gti. f = i=1 aigi. Thus, f(ag) = i=1 aigi(ag) = 0. Therefore, ag ∈ V (f1, . . . , fs). The other inclusion follows the same proof. ut

Deﬁnition 1.4.6. Let V ⊆ kn be an aﬃne variety, then

I(V ) = {f ∈ k[x1, . . . , xn]|f(a1, . . . , an) = 0, ∀(a1, . . . , an) ∈ V }

We call this the ideal of V .

n Lemma 1.4.7. If V ⊆ k be an aﬃne variety, then I(V ) ⊆ k[x1, . . . , xn] is an ideal.

Proof. It is clear 0 ∈ I(V ). If f, g ∈ I(V ), a = (a1, . . . , an) ∈ V , then (f + g)(a) = f(a) + g(a) = 0. Thus (f + g) ∈ I(V ). Let h ∈ k[x1, . . . , xn], then hf(a) = h(a)f(a) = h(a)0 = 0. Thus hf ∈ I(V ). Thus I(V ) is an ideal. ut

Then we look at two examples of I(V ).

Lemma 1.4.8. Let f1, . . . , fs ∈ k[x1, . . . , xn]. Then hf1, . . . , fsi ⊆ I(V (f1, . . . , fs)). The other inclusion doesn’t always work. Ps Proof. First, let a ∈ V (f1, . . . , fs). Let f ∈ hf1, . . . , fsi, then f = i=1 hifi, h ∈ k[x1, . . . , xn]. Ps Then f(a) = i=1 hifi(a) = 0. Thus f ∈ I(V (f1, . . . , fs)). The other side doesn’t work, consider counterexample V (x2, y2). ut

4 Proposition 1.4.9. Let V , W be affine varieties in kn. Then (i)V ⊆ W iff I(W ) ⊆ I(V ) (ii)V = W iff I(V ) = I(W ).

Proof. (i) Let V ⊆ W . Then if f vanishes on W , it vanishes in V . Then let I(W ) ⊆ I(V ). Thus, if g ∈ I(W ), then g ∈ I(V ), meaning that W has all common zeros. Thus V ⊆ W (ii) If V = W , then V ⊆ W , W ⊆ V . By (i) I(W ) = I(V ). The other direction follows. ut

1.5 Polynomial of One Variable

m Deﬁnition 1.5.1. Given a nonzero polynomial f ∈ k[x], let f = c0x + ... + cm, where m ci ∈ k, c0 6= 0, then we say c0x the leading term of f, m the degree of f. Proposition 1.5.2. (Division Algorithm) Let F be a ﬁeld, g a nonzero polynomial in F [x]. Then every f ∈ F [x] can be written as f = qg + r, where q, g ∈ F [x], and either r = 0 or deg(r) < deg(g). Furthermore, q, r are unique.

Proof. We ﬁrst prove the existence. If f = 0, then the case is trivial. If f 6= 0, we use induction. Let deg(f) = n. If n = 1, then f = ax + b, we have either g = c, then we can compute q, r and verify that r < g. Assume it is always the case for case n. Consider case n + 1. We can use long division to check this result. Now we prove uniqueness. Assume f = gq1 + r1, f = gq2 + r2. Then we have 0 = g(q1 − q2) + (r1 − r2). Now g(q1 + q2) is either 0 or has a degree greater than g. In the second case we have a contradiction. Therefore, q1 = q2, and r1 = r2. ut

Corollary 1.5.3. If F is a ﬁeld, f ∈ F [x] is a nonzero polynomial, then f has at most deg(f) roots in F .

Proof. We use induction. when deg(f) = 0, the solution is trivial. Assume it holds for deg(f) = n. Then consider n + 1 case. If a is a root of f, then f = q(x − a) + r by devision algorithm. But notice, f(a) = q(a)(a − a) + r = 0, thus r = 0. Thus, we observe that q is a degree n polynomial since (X − a) is degree one. By our hypothesis, there are at most n + 1 roots. we are done. ut

Deﬁnition 1.5.4. An ideal generated by one element is called a principal ideal. An integral domain with all ideals being principal ideals are called a principal ideal domain (PID).

Corollary 1.5.5. If F is a ﬁeld, then F [x] is a PID.

Proof. Consider I ⊆ F [x]. Choose f to be of minimum degree in I. If I = 0 this is trivial. Otherwise, if g, f ∈ I. Then g = qf + r. We know that q ∈ F [x], thus, qf ∈ I, r = g − qf ∈ I. Then r = 0. Therefore, g = qf, meaning that the ideal is generated by f. ut

Similarly, we have a similar deﬁnition for the gcd for many polynomials

Proposition 1.5.7. Let f, g ∈ k[x], (i)gcd(f, g) exists and is unique (ii)gcd(f, g) is a generator of the ideal (f, g) (iii) Euclidean Algorithm: gcd(f, g) = af + bg We can declare the same things with many polynomials.

Proof. Since k[x] is a PID, (f, g) = (h). Therefore, h|f, h|g. If p|f, p|g, then let f = ap, g = bp. Since h ∈ (f, g), h = cg + df = cbp + dap. Therefore, p|h. Thus h = gcd(f, g). Thus we proved uniqueness, (ii). If h0, h are both gcd(f, g), then h|h0 and h0|h. Thus h = h0, thus unique. Since (f, g) = (h), we have the Euclidean Algorithm. ut

6 Chapter 2

Notes: Grobner Bases

2.1 Introduction

In this chapter, we want to consider ways to ﬁnd all elements in an aﬃne variety. In a linear example, this is the reduced echelon form.

2.2 Orderings on Monomials

n Deﬁnition 2.2.1. A monomial ordering > on k[x1, . . . , xn] is a relation > on Z≥0 with the folloiwng properties: n (i)> is a total (or linear) ordering on Z≥0 n (ii)If α > β and γ ∈ Z≥0, then α + γ > β + γ. n n (iii)> is a well-ordering on Z≥0. (Every nonempty subset of Z≥0 has a smallest element under >) Given a monomial ordering >, we say that α > β or α = β.

n Lemma 2.2.2. An order relation > on Z≥0 is a well-ordering iff every strictly decreasing n sequence in Z≥0: α(1) > α(2) ... eventually terminates. Proof. Consider the contrapositive: > is not a well-ordering iff there is an infinite strictly n decreasing sequence in Z≥0. Thus, we claim is obvious. n (⇒)If > is not a well-ordering, then if A ⊂ Z≥0 non-empty. Pick α(1), by definition, there is a number α(2) less than α(1). Continue in this fashion, we get a infinite decreasing sequence. (⇐) By definition, it is not a well-ordering. ut

Deﬁnition 2.2.3. (Lexicographic Order) Let α = (α1, . . . , αn) and β = (β1, . . . , βn) be in n n Z≥0. We say α >lex β if the leftmost nonzero entry of the vector diﬀerence α − β ∈ Z is α β positive. We write x >lex x if α >lex β. n Proposition 2.2.4. The lex ordering on Z≥0 is a monomial ordering.

7 n Proof. (i) Since the order on Z≥0 is a total ordering, we see that >lex is a total ordering. (ii)If α >lex β, then we have αi − βi > 0. Then (α + γ) − (β − γ) has the nonzero entry αi − βi > 0 (iii) We prove by contradiction. Assume it is not a well-ordering, then there is a infinite strictly descending sequence α(1) >lex α(2) ... of elements α(1) > α(2) .... Consider first n α(i) ∈ Z≥0, then their first entries form a nonincreasing sequence of nonnegative integers. n Since Z≥0 is well-ordered, this sequence has to terminate in the first entries, say at α(l). Then consider elements after α(l), they have to form nonincreasing sequences in the later entries. Continue in this fashion, assume it all terminates at α(m), then we have α(m) = α(m + 1), contradiciton. ut

n Definition 2.2.5. (Graded Lex Order) Let α, β ∈ Z≥0. We say α >grlex β if |α| = Pn Pn i=1 αi > |β| = i=1 βi, or |α| = |β| and α >lex β. n Definition 2.2.6. (Graded Reverse Lex Order) Let α, β ∈ Z≥0. We say α >grevlex β if Pn Pn n |α| = i=1 αi > |β| = i=1 βi, or |α| = |β| and the rightmost nonzero entry of α − β ∈ Z is negative. P α Definition 2.2.7. Let f = α aαx be a nonzero polynomial in k[x1, . . . , xn], and > a n monomial order. The multidegree of f is max{α ∈ Z≥0|aα 6= 0}. The leading coefficient of multideg(f) f is amultideg(f) ∈ k, and the leading monomial of f is x , and the leading term of f is LT (f) = LC(f) · LM(f)

2.3 A Division Algorithm in k[x1, . . . , xn]

We have a division algorithm for k[x1, . . . , xn] as well. n Theorem 2.3.1. (Division Algorithm) Let > be a monomial order on Z≥0, let F = (f1, . . . , fs) be an ordered s-tuple of polynomials in k[x1, . . . , xn]. Then every f ∈ k[x1, . . . , xn] can be written as f = q1f1 + ... + qsfs + r, where qi, r ∈ k[x1, . . . , xn], and r either 0 or a linear conbination with coeﬀcients in k, and not divisible by any leading terms of our fi. We call r a remainder of f divided by F , and ∀i, multideg(fi) > multideg(r).

2.4 Monomial Ideals and Dickson’s Lemma

Definition 2.4.1. An ideal I ⊆ k[x1, . . . , xn] is a monomial ideal if there is a subset n P α A ⊆ Z≥0 such that I consists of all polynomial which are finite sums of the form α∈A hαx , α where hα ∈ k[x1, . . . , xn]. In this case, we write I = hx |α ∈ Ai Lemma 2.4.2. Let I = hxα|α ∈ Ai be a monomial ideal. Then a monomial xβ ∈ I iff xα|xβ. Proof. (⇐) This is the definition. β β Ps α(i) (⇒) If x ∈ I, then x = i=1 hix , where hi ∈ k[x1, . . . , xn], and α(i) ∈ A. If we expand this equation, we can see that xα|xβ. ut

8 Lemma 2.4.3. Let I be a monomial ideal, and let f ∈ k[x1, . . . , xn]. The following are equivalent: (i)f ∈ I (ii)Every term of f lies in I (iii) f is a k-linear combination of the monomials in I.

Corollary 2.4.4. Two monomial ideals are the same iﬀ they contain the same monomials.

α Theorem 2.4.5. (Dickson’s Lemma) Let I = hx |α ∈ Ai ⊆ k[x1, . . . , xn] be a monomial ideal. Then I has a ﬁnite basis.

α Proof. We prove by induction. Base case: n = 1, then I is generated by x1 . Let β be α β β minimal in A, then β < α, ∀α ∈ A. Thus, x |x . Thus I = hx1 i. Assume it works for n−1. Consider monomials in k[x1, . . . , xn−1, y]. Let J ⊆ k[x1, . . . , xn−1] be an ideal, and I = xαym, where xα ∈ J. Since J is a monomial, it is ﬁnitely generated. Thus, we can consider k[x1, . . . , xn−1, y] = k[x1, . . . , xn−1][y], thus, by base case, I = J[y]. Thus, I is also ﬁnitely generated. ut

n Corollary 2.4.6. Let > be a relation on Z≥0 satifying: n (i) > is a total ordering on Z≥0 n (ii) if α > β, and γ ∈ Z≥0, then α + γ > β + γ n Then > is well ordering iﬀ α ≥ 0, ∀α ∈ Z≥0. The proof is easy.

α(1) α(s) Proposition 2.4.7. A monomial ideal I ∈ k[x1, . . . , xn] has a basis x , . . . , x with the property that xα(i) does not divide xα(j) for i 6= j. This basis is unique and is called the minimal basis of I.

Proof. By Dickson’s Lemma, I has a ﬁnite basis of monomials. If one devides another, we ignore it. We repeat until we get a basis described. Assume another basis. Then we observe that xα(i)|xβ(j)|xalpha(1) which is a contradiction. Or we have to have i = 1, thus xβ(j) = xalpha(1), which means that this basis is unique ut

2.5 The Hilbert Basis Theorem and Grobner Bases

We denote by hLT (I)i the ideal generated by the elements of LT (I) (The set of leading terms of nonzero elements of I ⊆ k[x1, . . . , xn]). Notice that hLT (I)i can be larger than hLT (f1), . . . , LT (fs)i.

Proposition 2.5.1. Let I ⊆ k[x1, . . . , xn] be a non-zero ideal (i) hLT (I)i is a monomial ideal. (ii) There are g1, . . . , gt ∈ I such that hLT (I)i = hLT (g1), . . . , LT (gt)i.

9 Proof. (i) The leading monomial generates the leading term ideal. (ii) Since hLT (I)i is a monomial ideal, by Dickson’s lemma, it is generated by ﬁnitely many basis. Therefore, hLT (I)i = hLM(g1), . . . , LM(gt)i. Since each leading monomial diﬀers from the leading term by a constant, we have hLT (I)i = hLT (g1), . . . , LT (gt)i. ut

Theorem 2.5.2. (Hilbert Basis Theorem) Every ideal I ⊆ k[x1, . . . , xn] has a finite generating set. Proof. If I is trivial, then clearly it has a finite basis. Assume I is not trivial. We first put an order topology on I, and we find a leading term. Then I has an ideal of leading terms. Thus, we have hLT (I)i = hLT (g1), . . . , LT (gt)i. We want to show that I = hg1, . . . , gti. hg1, . . . , gti ⊆ I is clear. Now, if f ∈ I, then by division algorithm, f = q1g1 + ... + qtgt + r, where no term of r is divisible by any of the leading terms. Now notice that r = f − q1g1 − ... − qngn ∈ I, we know r must be 0. Thus f ∈ hg1, . . . , gti. Thus we have the other inclusion. ut

Definition 2.5.3. Consider k[x1, . . . , xn] with an order topology. A finite subset G = {g1, . . . , gt} of an ideal I ⊆ k[x1, . . . , xn] different from 0 is said to be a Grobner basis (standard basis) if hLT (I)i = hLT (g1), . . . , LT (gt)i.

Corollary 2.5.4. Fix an order topology. Every ideal I ⊆ k[x1, . . . , xn] has a Grobner basis. Furthermore, any Grobner basis for an ideal I is a basis of I. Deﬁnition 2.5.5. A ring R is called Noetherian if its ideals are generated by ﬁnitely many elements

Deﬁnition 2.5.6. The ascending chain condition of ideals: I1 ⊆ I2 ⊆ I3 ... forms a ﬁnite ascending chain (∃) an N ≥ 1 such that IN = IN+1 = ....

Theorem 2.5.7. k[x1, . . . , xn] satisﬁes the accending chain condition. ∞ Proof. Given an ascending chain I1 ⊆ I2 ⊆ I3 ..., consider I = ∪i=1Ii. We realize that I is an ideal in k[x1, . . . , xn]. By Hilbert Basis Theorem, I must have a ﬁnite generating set: I = hf1, . . . , fsi. But each of them need to be in one of the Ii. Thus we take N = max{i}. Hence, the chain stables at N. ut

Remark 2.5.8. R be a ring, it is Noetherian iﬀ it satisﬁes the ascending chain condition.

Deﬁnition 2.5.9. Let I ⊆ k[x1, . . . , xn] be an ideal. We denote V (I) = {(a1, . . . , an) ∈ n k |f(a1, . . . , an) = 0, ∀f ∈ I}

Proposition 2.5.10. V (I) be an aﬃne variety. If I = hf1, . . . , fsi, then V (I) = V (f1, . . . , fs).

Proof. By Hilbert Basis Theorem, I = hf1, . . . , fsi. Since fi ∈ I, if f(a1, . . . , an) = 0, ∀f ∈ I, then fi(a1, . . . , an) = 0. Thus V (I) ⊆ V (f1, . . . , fs). If on the other hand, we can write them as a linear combination. thus the other inclusion is satisﬁed. ut

We conclude that varieties are determined by ideals.

10 2.6 Properties of Grobner Bases

Proposition 2.6.1. Let I ⊆ k[x1, . . . , xn] be an ideal and let G = {g1, . . . , gt} be a Grobner basis for I. Then given f ∈ k[x1, . . . , xn], there is a unique r ∈ k[x1, . . . , xn] with the following: (i)No term of r is divisible by any of LT (g1), . . . , LT (gt). (ii) There is g ∈ I such that f = g + r Where r is the remainder on division of f by G.

Proof. By division algorithem, f = q1g1 + ... + qtgt + r, where r satisﬁes (i). By letting g = q1g1 + ... + qtgt we satisﬁes (ii). Now we need to prove uniqueness. Let f = g + r = g0 + r0. Then r − r0 = g − g0 ∈ I. If r 6= r0, LT (r − r0) ∈ hLT (I)i = 0 hLT (g1), . . . , LT (gt)i. Thus r − r is divisible by some LT (gi), contradiction. Thus uniqueness. ut

Deﬁnition 2.6.2. The remainder r above is called the normal form of f.

Corollary 2.6.3. Let G = {g1, . . . , gt} be a Grobner basis for an ideal I ⊆ k[x1, . . . , xn], let f ∈ k[x1, . . . , xn]. Then f ∈ I iﬀ r on division of f by G is 0. F Deﬁnition 2.6.4. We will write f for the remainder on division of f by the ordered s-tuple F = (f1, . . . , fs). If F is a Grobner basis for hf1, . . . , fsi, then we can regard F as a set by prop 2.5.1 Now we are going to discuss how to tell if a basis is a Grobner basis.

Deﬁnition 2.6.5. Let f, g ∈ k[x1, . . . , xn] be nonzero polynomials. If multideg(f) = α, multideg(g) = β, then let γ = (γ1, . . . , γn), where γi = max(α1, βi) for each i. We call xγ the least common multiple of LM(f) and LM(g), written as xγ = lcm(LM(f), LM(g)). And the S-polynomial of f and g is the combination xγ xγ S(f, g) = · f − · g LT (f) LT (g) Ps n Ps Lemma 2.6.6. Suppose we have i=1 pi, where multideg(pi) = δ ∈ Z≥0. If multideg( i=1 pi) < δ, then it is a linear combination of the S-polynomials S(pj, pl) for 1 ≤ j, l ≤ s, with each S(pj, pl) has multidegree < δ δ Ps Proof. Let di = LC(pi), so that dix is the leading term of pi. Since i=1 pi has a smaller Ps degree, we see that i=1 di = 0. Since pi and pj have the same LM, the S-polynomial S(p , p ) = 1 p − 1 p . Therefore Ps−1 d S(p , p ) = p +...+p − 1 (d +...+d p = i j di i dj j i=1 1 i s 1 s−1 ds ! s−1 s p1 + ... + ps−1 + ps. Thus the sum is a sum of S-polynomials, and it has multidegree < δ. ut

Theorem 2.6.7. (Buchberger’s Criterion) Let I be a polynomial ideal. Then a basis G = {g1, . . . , gt} of I is a Grobner basis of I iﬀ ∀i 6= j, the remainder on division of S(gi, gj) by G is zero.

11 Proof. (⇒) By the former Corollary. Pt (⇐) Let f ∈ I nonzero. f = i=1 higi, hi ∈ k[x1, . . . , xn]. Thus we know multideg(f) ≤ max(multideg(higi)) for higi 6= 0. Let δ = max(multideg(higi)) be minimal. Thus, multideg(f) ≤ δ. If we take equal sign, then LT (f) ∈ hg1 . . . , gti. Thus we are done. Assume not equal. Then we decrease δ by using the remainder equals to 0. ut

2.7 Buchberger;s Algorithm

Theorem 2.7.1. Let I = hf1, . . . , fsi= 6 0 by a polynomial ideal. Then a Grobner basis for I can be constructed in a ﬁnite number of steps in an algorithm. (Basically ﬁnd the remainder of the S-polynomial and if it is 0, the algorithm terminates, or we add the remainder into the basis and do it again)

We then talk about examples. But with this theorem, usually the basis computed are bigger than necessary

Lemma 2.7.2. Let G be a Grobner basis of I ⊆ k[x1, . . . , xn]. Let p ∈ G be a polynomial such that LT (p) ∈ hLT (G − {p})i. Then G − {p} is also a Grobner basis for I.

Now the same example but reduced to minimal Grobner basis.

Deﬁnition 2.7.3. A reduced Grobner basis for a polynomial ideal I is a Grobner basis G for I such that (i) LC(p) = 1, ∀p ∈ G (ii) ∀p ∈ G, no monomial of p lies in hLT (G − {p})i

Theorem 2.7.4. Let I 6= {0} be a polynomial ideal. Then, for a given monomial ordering, I has a reduced Grobner basis, and the reduced Grobner basis is unique.

Notice there is a connect between Buchberger’s Algorithm and Gauss elimination.

2.8 First Applications of Grobner Bases

We see examples in solving the ideal membership problem, the problem of solving polynomial equations, and the implicitization problem

12 Bibliography

[1] Cox, Little, O’Shea, Ideals, Varieties and Algorithms New York: Springer- Verlag, 2014