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**Section 4-2 Radical Ideals and the Ideal-Variety Correspondence by Pablo Spivakovsky-Gonzalez**

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**Let V be a variety. If fm 2 I(V), then f 2 I(V).**

-We ended Section 4-1 with Hilbert’s Nullstellensatz -In this section we will look at Hilbert’s Nullstellensatz from a different perspective -If we have some variety V, can we identify those ideals that consist of all polynomials which vanish on that variety? Lemma 1: Let V be a variety. If fm 2 I(V), then f 2 I(V).

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Proof: Let x 2 V. If fm 2 I(V), then (f(x))m = 0. This can only be true if f(x) = 0. This reasoning applies to any x 2 V, so we conclude that f 2 I(V). -Therefore, I(V) has the property that if some power of a polynomial is in the ideal, then that polynomial itself must also belong to I(V). -This leads to the definition of radical ideal.

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**{f : fm 2 I for some integer m ¸ 1 }.**

Definition 2: An ideal I is radical if fm 2 I for some integer m ¸ implies that f 2 I. -We can now rephrase Lemma 1 using radical ideals. Corollary 3: I(V) is a radical ideal. Definition 4: Let I ½ k[x1,…,xn] be an ideal. The radical of I, denoted , is the set {f : fm 2 I for some integer m ¸ 1 }.

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**1. I ½ , since f 2 I means f1 2 I and therefore f 2 by definition.**

Properties of : 1. I ½ , since f 2 I means f1 2 I and therefore f by definition. An ideal I is radical if and only if I = For any ideal I, is always an ideal. Example: Consider the ideal J = h x2, y3 i½ k[x, y]. Neither x nor y lie in J; but x and y

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**by the Binomial Theorem.**

Also, (x ¢ y)2 = x2y2 2 J, because x2 2 J. Then x ¢ y Finally, x + y To see this, we note that (x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4 , by the Binomial Theorem. Since each term above is a multiple of either x2 or y3, (x + y)4 2 J, and therefore x + y

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**Now consider the binomial expansion of (f + g)m+l-1 .**

Lemma 5: If I is an ideal in k[x1,…,xn] then is an ideal in k[x1,…,xn] containing I. Furthermore, is a radical ideal. Proof: I ½ has already been shown. We want to prove is an ideal. Let f, g ; then by definition there exist m, l 2 Z+ so that fm, gl 2 I. Now consider the binomial expansion of (f + g)m+l-1 .

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**Finally, suppose f 2 and h 2 k[x1,…,xn].**

Every term in the expansion has a factor figj with i + j = m + l – 1. Therefore, either i ¸ m or j ¸ l, so either fi 2 I or gj 2 I. This implies that figj 2 I, so every term of the expansion lies in I. Therefore, (f + g)m+l-1 2 I, so f + g Finally, suppose f and h 2 k[x1,…,xn]. This means that fm 2 I for some integer m ¸ l.

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**Theorem 6 (The Strong Nullstellensatz): **

Therefore, hmfm 2 I, or (h¢f)m 2 I, so hf This completes the proof that is an ideal. The book leaves the proof that is a radical ideal as an exercise at the end of the section. Theorem 6 (The Strong Nullstellensatz): Let k be an algebraically closed field. If I is an ideal in k[x1,…,xn], then I(V(I)) =

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**This means that f 2 . And because f was arbitrary,**

Proof: Consider any f Then by definition fm 2 I for some m ¸ l. Therefore fm vanishes on V(I), so clearly f must vanish on V(I) also. It follows that f 2 I(V(I)) , so we have ½ I(V(I)) . Conversely, suppose that f 2 I(V(I)). Then f vanishes on V(I). Now, by Hilbert’s Nullstellensatz, 9 m ¸ l such that fm 2 I. This means that f And because f was arbitrary,

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I(V(I)) ½ Before we had ½ I(V(I)), so clearly I(V(I)) = This concludes our proof. - Note: From now on, Theorem 6 will be referred to simply as “the Nullstellensatz”. - The Nullstellensatz allows us to set up a “dictionary” between algebra and geometry => very important!

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**Theorem 7 (The Ideal-Variety Correspondence): **

Let k be an arbitrary field. 1. The maps I affine varieties ===> ideals V ideals ===> affine varieties are inclusion-reversing. If I1 ½ I2 are ideals, then V(I1) ¾ V(I2) and, similarly, if V1 ½ V2 are varieties, then I(V1) ¾ I(V2).

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**In addition, for any variety V, we have V(I(V)) = V,**

so that I is always one-to-one. 2. If k is algebraically closed, and we restrict ourselves to radical ideals, then the maps I affine varieties ===> radical ideals V radical ideals ===> affine varieties are inclusion-reversing bijections which are inverses of each other.

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**By definition, every f 2 I(V) vanishes on V, so**

Proof: 1. The proof that I and V are inclusion reversing is given as an exercise at the end of the section. We will now prove that V(I(V)) = V, when V = V(f1,…,fs) is a subvariety of kn. By definition, every f 2 I(V) vanishes on V, so V ½ V(I(V)).

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**definition of I. Therefore, h f1,…,fs i ½ I(V).**

On the other hand, we have f1,…,fs 2 I(V) from definition of I. Therefore, h f1,…,fs i ½ I(V). V is inclusion reversing, hence V(I(V)) ½ V(h f1,…,fs i) = V. Before we had V ½ V(I(V)), and now we showed V(I(V)) ½ V.

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**We also know that V(I(V)) = V from Part 1.**

Therefore V(I(V)) = V, and I is one-to-one because it has a left inverse. This completes the proof of Part 1 of Thm. 7. 2. By Corollary 3, I(V) is a radical ideal. We also know that V(I(V)) = V from Part 1. The next step is to prove I(V(I)) = I whenever I is a radical ideal.

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**The Nullstellensatz tells us I(V(I)) = .**

Also, if I is radical, I = (Exercise 4). Therefore, I(V(I)) = I whenever I is a radical ideal. We see that V and I are inverses of each other. V and I define bijections between the set of radical ideals and affine varieties. This completes the proof.

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**Consequences of Theorem 7:**

- Allows us to consider a question about varieties (geometry) as an algebraic question about radical ideals, and viceversa. - We can move between algebra and geometry => powerful tool for solving many problems! - Note that the field we are working over must be algebraically closed in order to apply Theorem 7.

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**Questions about Radical Ideals:**

Consider an ideal I = h f1,…,fs i : 1. Radical Generators: Is there an algorithm to produce a set {g1,…,gm} so that = h g1,…,gm i ? 2. Radical Ideal: Is there an algorithm to determine if I is radical? 3. Radical Membership: Given f 2 k[x1,…,xn], is there an algorithm to determine if f ?

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**Proposition 8 (Radical Membership):**

Answer: -Yes, algorithms exist for all 3 problems. -We will focus on the easiest question, #3, the Radical Membership Problem. Proposition 8 (Radical Membership): Let k be an arbitrary field and let I = h f1,…,fs i ½ k[x1,…,xn] be an ideal. Then f 2 if and only if the constant polynomial 1 belongs to the ideal = h f1,…,fs, 1 – yf i ½ k[x1,…,xn,y].

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**Suppose 1 2 . Then we can write 1 as: s **

In other words, f 2 if and only if = k[x1,…,xn,y]. Proof: Suppose Then we can write 1 as: s 1 = pi (x1,…,xn, y) fi + q(x1,…,xn, y)(1 – yf), i=1 for some pi , q 2 k[x1,…,xn, y].

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**Now we multiply both sides by fm : fm = Ai fi , **

We set y = 1 / f(x1,…,xn). Then our expression becomes s 1 = pi (x1,…,xn, 1/f ) fi , i=1 Now we multiply both sides by fm : fm = Ai fi , for some polynomials Ai 2 k[x1,…,xn].

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**= ymfm + (1 – yf)(1 + yf +…+ ym-1fm-1 ) 2**

Therefore, fm 2 I, and so f Going the other way, suppose that f Then fm 2 I ½ for some m. At the same time, 1 – yf Then, 1 = ymfm + (1 – ymfm ) = = ymfm + (1 – yf)(1 + yf +…+ ym-1fm-1 ) 2

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**so the proof is complete.**

Hence, f implies that And before we had that implies f 2 so the proof is complete. Radical Membership Algorithm: -To determine if f ½ k[x1,…,xn] we first compute a reduced Groebner basis for:

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**h f1,…,fs, 1 – yf i ½ k[x1,…,xn,y]. -If the result is {1}, then f 2 .**

-Example: Consider the ideal I = h xy2 + 2y2, x4 – 2x2 + 1 i in k[x, y]. We want to determine if f = y – x lies in Using lex order on k[x, y, z], we compute a reduced Groebner basis of the following ideal:

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**The basis we obtain is {1}, so by Proposition 8 f 2 .**

= h xy2 + 2y2, x4 – 2x2 + 1, 1 – z (y – x2 + 1) i The basis we obtain is {1}, so by Proposition 8 f - In fact, (y – x2 + 1)3 2 I, but no lower power of f is in I. Principal Ideals: -If I = h f i, we can compute the radical of I as follows:

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**Proposition 9: Definition 10:**

Let f 2 k[x1,…,xn] and I = h f i. If is the factorization of f into a product of distinct irreducible polynomials, then = h f1f2 ··· fr i Definition 10: If f 2 k[x1,…,xn] is a polynomial, we define the reduction of f, denoted fred , to be the polynomial such that h fred i = , where I = h f i.

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**-A polynomial is said to be reduced, or square-free, if**

f = fred . -Section 4-2 ends with a formula for computing the radical of a principal ideal => see pg. 181.

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**Stay tuned for the next lecture,**

Sources Used - Ideals, Varieties, and Algorithms, by Cox, Little, O’Shea; UTM Springer, 3rd Ed., 2007. Thank You! Stay tuned for the next lecture, by ShinnYih Huang!

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