# Polynomials, Ideals, and Gröbner Bases

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1 Polynomials, Ideals, and Gröbner Bases Notes by Bernd Sturmfels for the lecture on April 10, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra We fix a field K. Some examples of fields are the rational numbers Q, the real numbers R, the complex numbers C, the finite field F q with q elements, or the field Q p of p-adic numbers. The last three fields are not algebraically closed, and we may consider their algebraic closures. Those fields are denoted Q, F q, Q p. Other fields occurring in this course are the rational functions Q(t), along with its algebraic closure, and Puiseux series C{{t}} = C{{t}}. The set of all polynomials in n variables x 1, x 2,..., x n with coefficients in our field K is denoted K[x] = K[x 1, x 2,..., x n ]. This is a commutative ring. If the number n is small then we typically use letters without indices to denote the variables. For instance, we often write K[x], K[x, y], or K[x, y, z] for the polynomial ring when n 3. The polynomial ring K[x] is an infinite-dimensional K-vector space. A distinguished basis is given by the monomials x a = x a 1 1 x a 2 2 x an n, one for each nonnegative integer vector a = (a 1, a 2,..., a n ) N n. Thus every polynomial f K[x] is written uniquely as a finite sum f = a c a x a. The degree of the polynomial f is the maximum of the quantities a = a 1 + a a n, where c a 0. Polynomials of degree 1, 2, 3, 4 are called linear, quadratic, cubic, quartic. Figure 1: A cubic surface with four singular points. 1

2 For example, the following is a polynomial of degree 3 in n = 3 variables: 1 x y f = det x 1 z = 2xyz x 2 y 2 z (1) y z 1 The zero set of the polynomial f is the surface in R 3 that is shown in Figure 1. This determinantal surface consists of all points at which the rank of the 3 3-matrix in (1) drops. We note that our cubic surface has four singular points, namely the points (1, 1, 1), (1, 1, 1), ( 1, 1, 1), and ( 1, 1, 1). These four points are the common zeros in R 3 of the cubic f and its three partial derivatives of f: f x = 2yz 2x, f y = 2xz 2y, f z = 2xy 2z. Equivalently, they are the points at which the rank of the 3 3-matrix in (1) drops to 1. Definition 1. An ideal is a subset I of the polynomial ring K[x] such that (a) if f K[x] and g I then fg I; (b) if g, h I then g + h I. Equivalently, I is closed under taking linear combination with polynomial coefficients. Given any subset F of K[x], we write F for the smallest ideal containing F. This is the ideal generated by F. It is the set of a polynomial linear combinations of finite subsets of F. Proposition 2. If I and J are ideals in K[x] then the following subsets are ideals as well: the sum I + J, the intersection I J, the product IJ and the quotient (I : J). The latter two are defined as follows: IJ = fg : f I, g J and (I : J) = { f K[x] : fj I }. Proof. The product IJ is an ideal by definition. For the others one checks (a) and (b). We shall carry out this check for the ideal quotient (I : J). To show (a), suppose that f K[x] and g (I : J). There exists h J such that gh I. Then f(gh) I and hence h(fg) I, so that fg (I : J). For (b), suppose g, h (I : J). There are u, v J so that ug I and vh I. This implies uv J, uvg I and uvh I, hence uvg + uvh = uv(g + h) I, and g + h (I : J). We have shown that (I : J) is an ideal. The Euclidean algorithm works in the polynomial ring K[x] in one variable. This implies that K[x] is a principal ideal domain (PID), i.e. every ideal I is generated by one element. That generator can be uniquely factored into irreducible polynomials. Every polynomial ring K[x] is a unique factorization domain (UFD). However, K[x] is not a PID when n 2. Example 3 (n = 1). Consider the following two ideals in Q[x]: I = x 3 + 6x x + 8 and J = x 2 + x 2. 2

3 We wish to compute the four new ideals in Proposition 2. For this, it helps to factor: We then find I = (x + 2) 3 and J = (x 1)(x + 2). I J = (x 1)(x + 2) 3 IJ = (x 1)(x + 2) 4 I + J = x + 2 I : J = (x + 2) 2. We conclude that arithmetic in Q[x] is just like arithmetic in the ring of integers Z. Ideals in a commutative ring play the same role as normal subgroups in a group. These are the subobjects that can be used to define quotients. Consider the quotient of K-vector spaces K[x]/I. Its elements are the congruence classes f + I modulo the subvectorspace I. The ideal axioms (a) and (b) in Definition 1 ensure that the following identities hold: (f + I) + (g + I) = (f + g) + I and (f + I)(g + I) = fg + I. (2) Corollary 4. If I K[x] is an ideal then K[x]/I is a ring. We call this the quotient ring. Properties of the ideal I correspond to properties of the quotient ring K[x]/I, as follows: property of the ideal definition property of the quotient ring I is maximal f I f invertible mod I K[x]/I is a field I is prime fg I f I or g I K[x]/I is an integral domain I is radical ( s : f s I) f I there are no nilpotent elements I is primary fg I and g I ( s : f s I) every zerodivisor is nilpotent Example 5. The ideal I = x x + 34, 3y 2x 13 is maximal in R[x, y]. The field R[x, y]/i is isomorphic to the complex numbers C = R[i]/ i One isomorphism is gotten by sending i = 1 1 to 169 (x + 5y). The square of that expression is 1 mod I. Examples for the other three classes of ideals are given in the next proof. Proposition 6. We have the following implications: I maximal I prime I radical, I primary. None of these implications is reversible. Every intersection of prime ideals is radical. Proof. The first implication holds because there are no zerodivisors in a field. Take g = f s 1 to see that prime implies radical. Prime implies primary is clear. To see that no implication is reversible consider the following three ideals in the polynomial ring R[x, y] with n = 2: I = x 2 is primary but not radical, I = x(x 1) is radical but not primary, I = x is prime but not maximal. The last statement holds because every intersection of radical ideals is a radical ideal. 3

4 We now revisit the singular surface in Figure 1 from the perspective of ideals. Example 7 (n = 3). Consider the ideal generated by the partial derivatives of the cubic f: I = f x, f y, f = xy z, xz y, yz x R[x, y, z]. x The given cubic f is not in this ideal because any polynomial in I has zero constant term. The ideal I is radical becasue we can write it as the intersection of five maximal ideals: I = x, y, z x 1, y 1, z 1 x 1, y+1, z+1 x+1, y 1, z+1 x+1, y+1, z 1. The Chinese Remainder Theorem implies that the quotient ring is a product of fields: R[x, y]/i R R R R R. The cubic f lies in the last four maximal ideals. Their intersection is equal to I + f. The zero set of the radical ideal I + f consists of the four singular points on the surface. Every ideal has many different generating sets, and there is no canonical notion of basis. For instance, the set F = {x 6 1, x 10 1, x 15 1} minimally generates the ideal x 1. Of course, the singleton {x 1} is a preferable generating set, given that every ideal in K[x] is principal for n = 1. The Euclidean algorithm transforms the set F into the set {x 1}. A certificate for the fact that x 1 lies in the ideal generated by F is the identity x 5 (x 6 1) (x 5 + x) (x 10 1) + 1 (x 15 1) = x 1. Such certificates can be found with the Extended Euclidean Algorithm. Finding certificates for ideal membership when n 2 comes up when we discuss the Nullstellensatz in Lecture 5. Gaussian elimination carries out a similar transformation for ideals that are generated by linear polynomials. For example, the following two ideals in Q[x, y, z] are identical: 2x + 3y + 5z + 7, 11x + 13y + 17z + 19, 23x + 29y + 31z + 37 = 7x 16, 7y + 12, 7z + 9. Undergraduate linear algebra taught us how to transform the three generators on the left into the simpler ones on the right. This is the process of solving a system of linear equations. We next introduce Gröbner bases. This theory offers a method for computing with ideals in a polynomial ring K[x]. Implementations of Gröbner bases are available in all major computer algebra systems, and we strongly encourage our readers to experiment with this. Informally, we can think of Gröbner bases as a version of the Euclidean algorithm for polynomials in n 2 variables, and as version of Gaussian elimination for polynomials of degree 2. Gröbner bases for ideals in K[x] are fundamental in non-linear algebra, just like Gaussian elimination for matrices were fundamental when we studied linear algebra. We identify the set N n with the monomial basis of K[x]. The coordinatewise order on N n corresponds to divisibility of monomials, i.e. we have a b if and only if x a divides x b. Theorem 8 (Dickson s Lemma). Any infinite subset of N n contains a pair satisfying a b. 4

5 Proof. We proceed by induction on n. The statement is trivial for n = 1. Any subset of cardinality at least two in N contains a comparable pair. Suppose now that Dickson s Lemma has been proved for n 1, and consider an infinite subset M of N n. For each i N let M i denote the set of all vectors a N n 1 such that (a, i) M. If some M i is infinite then we are done by induction. Hence each M i is finite, and we have M i for infinitely many i. The infinite subset i=0m i of N n 1 satisfies the assertion. This means that its subset of minimal elements with respect to the coordinatewise order is finite. Hence there exists an index j such that all minimal elements are contained in the finite set j i=0 M i. Pick any element (b, k) M k for k > j. Since b is not minimal in i=0m i, there exists an index i with i j < k and an element a M i with a b. Then we have (a, i) (b, k) in M. Corollary 9. For any set M N n, its subset of coordinatewise minimal elements is finite. Definition 10. A total ordering on N n is a monomial order if, for all a, b, c N n we have (0, 0,..., 0) a; a b implies a + c b + c. This gives a total order on all monomials in K[x]. Three standard examples are: the lexicographic ordering: a lex b if the leftmost non-zero entry of b a is positive. the degree lexicographic order: we set a deglex b if either a < b, or a = b and the leftmost non-zero entry of b a is positive. the degree reverse lexicographic order: we set a revlex b if either a < b, or a = b and the rightmost non-zero entry of b a is negative. All three orders satisfy x 1 x 2 x n, but they differ on monomials of higher degree. Throughout this course we specify a monomial order by giving the name of the order and how the variables are sorted. For instance, we might say: let denote the degree lexicographic order on K[x, y, z] given by y z x. Further choices of monomial orderings can be obtained by assigning positive weights to the variables. See [1, Exercise 11 in 2.4]. Remark 11. Fix a monomial order and let M be any infinite subset of N n. Then M has a unique minimal element with respect to. To see this, apply Dickson s Lemma. Our set M has a finite subset of minimal elements with respect to the component-wise partial order on N n. This finite subset is linearly ordered by, and we select its minimal element. We now fix a monomial order. Given any nonzero polynomial f K[x], its initial monomial in (f) is the -largest monomial x a among those that appear in f with non-zero coefficient. For a comparison among our orders, let n = 3 with variable order x y z: If f = x 2 + xz 2 + y 3 then in lex (f) = x 2, in deglex (f) = xz 2, and in revlex (f) = y 3. 5

6 For any ideal I K[x], we define the initial ideal of I with respect to as follows: in (I) = in (f) : f I. This is a monomial ideal, i.e. it is generated by a set of monomials. A priori, this generating set is infinite. However, it turns out that we can always choose a finite subset that suffices. Proposition 12. Fix a monomial order. Every ideal I has a finite subset G such that in (I) = in (f) : f G. Such a finite subset G of I is called a Gröbner basis for I with respect to. Proof. Suppose no such finite set G exists. Then we can create a list of infinitely many polynomials f 1, f 2, f 3,... in I such that none of the initial monomials in (f i ) divides any other initial monomial in (f j ). This would be a contradiction to Dickson s Lemma. We next show that every Gröbner basis actually generates its ideal. Theorem 13. If G is a Gröbner basis for an ideal I then I = G. Proof. Suppose that G does not generate I. Among all polynomials f in the set difference I\ G, there exists an f whose initial monomial x b = in (f) is minimal with respect to. This follows from Remark 11. Since x b in (I), there exists an element g G that divides x b, say x b = x c g. Now, f x c g is a polynomial with strictly smaller leading monomial. It lies in I but it does not lie in the ideal G. This is a contradiction to the choice of f. Corollary 14 (Hilbert s Basis Theorem). Every ideal I in K[x] is finitely generated. Proof. Fix any monomial order. By Proposition 12, the ideal I has a finite Gröbner basis G. By Theorem 13, I is generated by G. Gröbner bases are not unique. If G is a Gröbner basis of I with respect to then so is every other finite subset of I that contains G. In that sense, Gröbner bases differ from the bases we know from linear algebra. The issue of minimality and uniqueness is addressed next. Definition 15. Fix I and. A Gröbner basis G is reduced if the following conditions hold: (a) The leading coefficient of each polynomial g G is 1. (b) For any two distinct g, h G, no monomial occurring in g is a multiple of in (h). In what follows we fix an ideal I K[x] and a monomial ordering. Theorem 16. The ideal I has a unique reduced Gröbner basis with respect to. 6

7 Proof idea. We refer to [1, 2.7, Theorem 5]. The idea is as follows. We start with any Gröbner basis G and we turn it into a reduced Gröbner basis by the following steps. First we divide each g G by its leading coefficient to make it monic, so that (a) holds. We remove all elements g from G whose initial monomial is not a minimal generator of in (I). For any pair of polynomials with the same initial monomial we delete one of them. Next we apply the division algorithm [1, 2.3] to any trailing monomial until no more trailing monomial is divisible by any leading monomial. The resulting set is the reduced Gröbner basis. Let S (I) be the set of all monomials x b that are not in the initial ideal in (I). We call these x b the standard monomials of I with respect to. Theorem 17. The set S (I) of standard monomials is a basis for the K-vectorspace K[x]/I. Proof. The image of S (I) in K[x]/I is linearly independent because every non-zero polynomial f has at least one monomial, namely in (f), that is not in S. We next prove that S (I) span K[x]/I. Suppose not. Then there exists a monomial x c which is not in the K-span of S (I) modulo I. We may assume that x c is minimal with respect to the term order. Since x c is not in S (I), it lies in the initial ideal in (I). Hence there exists h I with in (h) = x c. Each monomial in h other than x c is smaller with respect to, so it lies in the K-span of S (I) modulo I. Hence x c has the same property. This is a contradiction. Software for Gröbner bases rests on the Buchberger Algorithm [1, S 2.7]. This is implemented in all major computer algebra systems. It takes as its input a monomial order and a finite set F of polynomials in K[x], and its output is the unique reduced Gröbner basis G for the ideal I = F with respect to. In what follows we present some examples of pairs (F, G) for n = 3. In each case we take the lexicographic term order with x y z. Example 18. The input F Q[x, y, z] is transformed into the reduced Gröbner basis G: For n = 1, the reduced Gröbner basis consists of the greatest common divisor: F = {x 3 6x 2 5x 14, 3x 3 +8x 2 +11x+10, 4x 4 +4x 3 +7x 2 x 2}, G = {x 2 +x+2}. For linear polynomials, running Buchberger s algorithm amounts to Gaussian elimination: For F = {2x + 3y + 5z + 7, 11x + 13y + 17z + 19, 23x + 29y + 31z + 37}, the reduced Gröbner basis G = {x 16, y + 12, z + 9 }. Leading monomials are underlined Here is another ideal we saw earlier: F = {xy z, xz y, yz x}, G = {x yz, y 2 z 2, yz 2 y, z 3 z}. There are precisely 5 standard monomials: S (I) = {1, y, z, yz, z 2 }. This is consistent with Example 7, where we saw that F has precisely 5 zeros in C 3. This input is a curve in the (y, z)-plane parametrized by two cubics in one variable x: F = {y x 3 + 4x, z x 3 x + 1}. The Gröbner basis reveals its implicit equation: G = { x y z 1 5, y3 3y 2 z 3y 2 + 3yz 2 + 6yz + 28y z 3 3z z + 99 }. Let z be the sum of x = 3 7 and y = 4 5. We write this as F = {x 3 7, y 4 5, z x y}. Then z = is algebraic of degree 12 over Q. Its minimal polynomial is seen in G = {z 12 28z 9 15z z z z z z z ,...}. 7

8 The elementary symmetric polynomials F = {x + y + z, xy + xz + yz, xyz} have the Gröbner basis G = { x + y + z, y 2 + yz + z 2, z 3 }. There are six standard monomials. The quotient Q[x, y, z]/i carries the regular representation of the symmetric group S 3. For these instances, what is the reduced Gröbner basis for the degree lexicographic order? In general, the choice of monomial order can make a huge difference in the complexity of the size of the reduced Gröbner basis, even if the input polynomials are homogeneous. Example 19 (Intersecting two quartic surfaces in P 3 ). Consider the ideal I generated by two random homogeneous polynomials of degree 4 in n = 4 variables. If is the degree reverse lexicographic order then the reduced Gröbner basis G contains 5 elements of degree up to 7. If is the lexicographic order then G contains 150 elements of degree up to 73. Naturally, one uses a computer to find the 150 elements in the aforementioned Gröbner basis. Many computer algebra systems offer an implementation of Buchberger s algorithm for Gröbner bases. Our readers are strongly encouraged to use a computer algebra system. Exercises 1. Show that the polynomial f = 5x 3 25x 2 y + 25y xy 50y 2 5x + 25y 1 is a product of three linear factors in R[x, y]. Draw the plane curve {f = 0}. 2. For n = 2, define a monomial ordering such that (2, 3) (4, 2) (1, 4). 3. Let n = 2 and consider the ideals I = x, y 2 and J = x 2, y. Compute I + J, I J, IJ and I 3 J 4 = IIIJJJJ. How many minimal generators does the ideal I 123 J 234 have? 4. The radical I of an ideal I is the smallest radical ideal containing I. Prove: The radical of a primary ideal is prime. The radical of a principal ideal is principal. The radical of a monomial ideal is a monomial ideal. 5. Show that the following inclusions always hold, and that they are strict in general: I J IJ and in ( I ) in (I). 6. Using Gröbner bases, find the minimal polynomials of and Find the implicit equation of the parametrized curve {(x 5 6, x 7 8) R 2 : x R}. 8. Investigate the ideal I = x 3 yz, y 3 xz, z 3 xy. Is it radical? If not, find I. Regarding I as a system of 3 equations in 3 unknowns, what are the solutions in R 3? 8

9 9. Find an ideal in Q[x, y] whose reduced Gröbner basis (for the lexicographic monomial order) consists of precisely 5 elements and there are precisely 19 standard monomials. 10. Prove: An ideal is principal if and only if its reduced Gröbner basis is a singleton. 11. Let I be the ideal generated by the n elementary symmetric polynomials in n variables. Pick a monomial ordering and determine the initial monomial ideal in (I). 12. Let X be 2 2-matrix whose n = 4 entries are variables. Let I s be the ideal generated by the entries of the matrix power X s for s = 2, 3, 4,.... Investigate these ideals. 13. A symmetric 3 3-matrix with unknown entries has seven principal minors: three of size 1 1, three of size 2 2, and one of size 3 3. Does there exist an algebraic relation among these determinants? Use lexicographic Gröbner bases to answer this question. 14. Prove that if in (I) is a radical ideal then I is a radical ideal. Does the converse hold? 15. Does the cubic surface in Figure 1 contain any straight line? Find all such lines. References [1] D. Cox, J. Little and D. O Shea: Ideals, Varieties, and Algorithms. An introduction to computational algebraic geometry and commutative algebra, Third edition, Undergraduate Texts in Mathematics, Springer, New York,

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