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*To*: fom@math.psu.edu*Subject*: FOM: 71:Ackerman/Algebraic Geometry/1*From*: Harvey Friedman <friedman@math.ohio-state.edu>*Date*: Fri, 10 Dec 1999 13:52:30 -0500*Sender*: owner-fom@math.psu.edu

*Manuscript with proofs is available upon request.* NOTE: This posting stops at the end of section 6. The next posting on this topic will include sections 7 - 9. This is a folowup to postings #40 and #43 about the Ackerman function in algebraic geometry. We have obtained more striking presences of Ackerman in algebraic geometry that imply the earlier results - at least as far as asymptotics are concerned. So this posting is entirely self contained. We start with two sections that discuss the new presences of Ackerman in algebraic geometry. We follow this by two sections on our earlier presences of Ackerman which, in the case of ideals, was extensively published on by A. Seidenberg. However, he obtained only very poor upper bounds and no lower bounds. We conclude with two sections on additional presences of Ackerman in algebraic geometry, the first of which has been known to us for many years. THE ACKERMAN FUNCTION IN ELEMENTARY ALGEBRAIC GEOMETRY by Harvey M. Friedman friedman@math.ohio-state.edu www.math.ohio-state.edu/~friedman/ December 10, 1999 abstract 1. Full subideals. 2. Algebraic approximations. 3. Upper bounds for algebraic approximations. 4. Ascending chains of ideals - historical notes. 5. Ascending chains of ideals. 6. Decreasing chains of algebraic sets. 7. Degrees in ideal bases. 8. Degrees in polynomial presentations of algebraic sets. 9. Additional formulations. The Ackerman hierarchy and Ackerman function is defined as follows. Let f:Z^+ into Z^+ be strictly increasing. Define f':Z^+ into Z^+ by f'(n) = f... f(1), where there are n f's. Define A_1:Z^+ into Z^+ as the doubling function; i.e., A_1(n) = 2n. For each k >= 1, define A_k+1 = A_k'. Finally, define A(k,n) = A_k(n), and A(k) = A(k,k). The former is the binary Ackerman function, and the latter is the unary Ackerman function. The following facts about A are useful, and are easily proved in the order stated. LEMMA. For all k,n >= 1, n < A_k(n) < A_k(n+1). For all k >= 1 and n >= 3, A_k(n) < A_k+1(n). For all k,n >= 1, A_k(n) <= A_k+1(n). For all k >= 1, A_k(1) = 2, A_k(2) = 4, and A_k(3) >= 2^k+1. For all k >= 3, A_k(3) >= A_k-2(2^k) > A_k-2(k-2) = A(k-2). As a function of k, A_k(3) eventually strictly dominates each A_n, n >= 1. Ackerman's original definition is similar but is ternary. The growth rates are the same. 1. FULL SUBIDEALS. An ideal in a commutative ring with unit is any subgroup which is closed under multiplication by any ring element. Thus the smallest ideal is {0}. The ideal {0} is generated by the empty set (and also by {0}). We say that J is a subideal of I if and only if J is an ideal that is included in I. The degree of an ideal in a polynomial ring is the least d such that the ideal has a set of generators all of which have degree at most d. Throughout this section, we fix a field F. Let k >= 0 and I be an ideal in the polynomial ring F[x_1,...,x_k]. For each n >= 0, we let I*n be the ideal generated by the elements of I of degree n. We let I*<=n be the ideal generated by the elements of I of degree <= n. For this purpose, we take an empty intersection to be all of F[x_1,...,x_k]. This basic construction can be studied in the much more general context of an arbitrary subset I of the polynomial ring. However, we do not disucss this here. We refer to the I*n as the full subideals of I. Note that the ideals {0} and I are full subideals of I provided I is not the whole polynomial ring. If I is the whole polynomial ring then its only full subideal is itself. THEOREM 1.1. For all n >= 0, I*n = I*<=n. If I*n has degree d then I*n = I*d. The full subideals of I form a chain under inclusion of length at most deg(I)+1, with at most one exact subideal of any given degree in the interval [0,deg(I)]. Note that there may be one or more missing full subideals of I. I.e., there may be at most deg(I) full subideals of I. We call I rich if and only if it has a full complement of full subideals - i.e., it has deg(I)+1 full subideals. THEOREM 1.2. I is rich if and only if for all 0 <= n <= deg(I), I*n has degree n. THEOREM 1.3. Every full subideal of every rich ideal is a rich ideal. The degrees of rich ideals in F[x_1,...,x_k] form an initial segment of the nonnegative integers. THEOREM 1.3. The degrees of the rich ideals of F[x_1,...,x_k] form a finite initial segment of the nonnegative integers. The union of these finite initial segments over all fields F is still finite. Upper bounds for Theorem 1.3 follow from upper bounds in section 5. Lower bounds for Theorem 1.3 are obtained by the following construction, even for monomial ideals in any F[x_1,...,x_k]. Let z_1,...,z_t be elements of N^k, where 1) for each i, the sum of the coordinates of z_i is i; 2) for all i < j, it is not the case that z_i <= z_j coordinatewise. For each z in N^k, let M(z) be the monomial x_1^z_1 x_2^z_2 ... x_k^z_k. Let I be the ideal in F[x_1,...,x_k] generated by M(z_1),...,M(z_t). LEMMA 1.4. The ideal I generated by M(z_1),...,M(z_t) is rich. In particular, for all 0 <= n <= t, I*n is the ideal generated by M(z_1),...,M(z_n). We had analyzed such sequences z_1,...,z_t from N^k already in the 1980's as a spinoff of our finite forms of Kruskal's theorem - and finite statements associated with well quasi orders. As k grows, the longest length of such sequences from N^k grows like Ackerman's function, eventually strictly dominating all primitive recursive functions, and being bounded by Ackerman's function at a constant multiple of the argument. Much more exact estimates can be obtained with additional work. 2. ALGEBRAIC APPROXIMATIONS. Ideals are fine for algebraists. But geometers and others may prefer algebraic sets, particularly in the reals or complexes. Let F be a field and k >= 1. An algebraic subset of F^k is the set of simultaneous zeros of a nonempty finite set of polynomials in k variables with coefficients from F. The degree of an algebraic set is the least d such that it is the set of simultaneous zeros of a nonempty finite set of polynomials in k variables each of which have degree at most d. ***NOTE: This is not the most common notion of degree for algebraic geometry. There is a close relationship between the degree concepts, and we will discuss what happens to our results when alternative definitions of degree are used - but in later postings. In particular, we already know that all positive results and Ackerman upper bounds hold for a very wide range of notions of degree including those used most commonly in algebraic geometry. The issue is one of lower bounds. Do our lower bound results hold with more sophisticated notions of degree?*** We say that an algebraic set E is defined by a set of polynomials if and only if it is the set of simultaneous zeros of that set of polynomials. Throughout this section, we fix an infinite field F. Let A be an algebraic subset of F^k. For each n >= 0, we let A*n be the least algebraic superset of A of degree n. We let A*<=n be the intersection of all algebraic supersets of A of degree <= n. For this purpose, we take an empty intersection to be all of F^k. We can apply this basic construction more generally to an arbitrary subset of F^k. However, we do not discuss this here. THEOREM 2.6. Let A be an algebraic subset of F^k, F infinite. For all n >= 0, A*n = A*<=n. THEROEM 2.7. Let A be an algebraic subset of F^k, F infinite. If A*n has degree d then A*n = A*d. The algebraic approximations of A form a chain under inclusion of length at most deg(A)+1, with at most one algebraic aproximation of any given degree in the interval [0,deg(A)]. Note that there may be one or more missing algebraic approximations of A. I.e., there may be at most deg(A) algebraic approximations of A. We call A rich if and only if it has a full complement of algebraic approximations - i.e., it has deg(A)+1 algebraic approximations. THEOREM 2.8. A is rich if and only if for all 0 <= n <= deg(A), A*n has degree n. THEOREM 2.9. Every algebraic approximation of every rich algebraic set is a rich algebraic set. The degrees of rich algebraic subsets of F^k form an initial segment of the nonnegative integers. THEOREM 2.10. The degrees of rich algebraic subsets of F^k form a finite initial segment of the nonnegative integers. The union of these finite initial segments over all fields F is still finite. 3. LOWER BOUNDS FOR ALGEBRAIC APPROXIMATIONS. Lower bounds for Theorem 2.10 require a new construction. An interesting aspect of these lower bounds is that we obtain them by considering finite sets only. Finite sets are automatically algebraic, but their degree is generally an intricate matter. THEOREM 3.13. Let k >= 1 and F be an infinite field. There is a rich finite subset of F^k+6 of degree A_k(k). THEOREM 3.14. Let k >= 1 and F be a sufficiently large finite field. There is a rich subset of F^k+6 of degree A_k(k). In fact, the finite field need only be a little bit larger than A_k(k), which we will eventually take the trouble to make precise. 4. ASCENDING CHAINS OF IDEALS - HISTORICAL NOTES. The results here about ascending chains of ideals were obtained in the 80's and discussed here in postings #40 and #43. Seidenberg had earlier intensively investigated the same statement about ascending chains of ideals in the following papers: A. Seidenberg, An elimination theory for differential algebra, Univ. Calif. Pubs. Math. 3 (1956), 31-65. A. Seidenberg, On the length of a Hilbert ascending chain, Proc. AMS, Vol. 29, No. 3, August 1971, 443-450. A. Seidenberg, Constructive proof of Hilbert's theorem on ascending chains, Trans. AMS, Vol. 174, December 1972, 305-312. The first paper is quoted in the last two and has a partial result. Actually, his is more general in that he considers arbitrary bounds on the degrees of the ideals. I had also dealt with this formulation but did not report it in #40 and #43. Of course, my work is all after Seidenberg. Seidenberg proves no lower bounds. Also, he states a multi recursive bound in each dimension k, rather than my primitive recursive bounds. He states only primitive recursive bounds in dimension <= 2, and states that primitive recursivity for dimension >= 3 is "doubtful." Also, Seidenberg does not consider corresponding statements about algebraic sets, which of course follows from these statements about ideals. Seidenberg also discusses these chains in A. Seidenberg, Survey of constructions in Noetherian rings, Univ. of Cal. Berkeley. A. Seidenberg, Constructions in algebra, Trans. AMS, Vol. 197, 1974, 273-313. I think he also disucces it in a fifth paper and probably others, entitled "What is Noetherian" but I haven't got a hold of a copy of that paper. Seidenberg's theorem can be viewed as a kind of finite form of the Hilbert basis theorem. He himself viewed it as a constructive form of the Hilbert basis theorem. **Our original finite forms of Kruskal's theorem in 1981-82 are to Kruskal's theorem as Sidenberg's theorem is to the Hilbert basis theorem. Bounds on the degrees become bounds on the number of vertices of the trees. It is interesting to note that Seidenberg and I had the same idea for getting finite forms in the two contexts - Hilbert's basis theorem and Kruskal's theorem - namely to look at finite sequences with bounds placed on the terms. ** Yet we searched hard for a good finite form of Kruskal's theorem invovlinga single tree rather than a sequence of trees. We were quite successful with this, and reported the results in posting #27, together with sketches of the proofs. After some experience with lecturing on this, my favorite is: 1) if T is a sufficiently tall thin tree, there exists 1 <= i < j <= hgt(T) and an inf preserving embedding from T[<=i] into T[<=j] which maps T[i] into T[j]. I.e., if T is a sufficiently tall tree of valence <= k, then there exists ... . It can be stated for arbitrary finite trees as follows. 2) for all k there exists n such that if T is any finite tree of valence <= k, there exists 1 <= i < j <= n and an inf preserving embedding from T[<=i] into T[<=j] which maps T[i] into T[j]. So it makes sense to search for theorems about a single ideal. We have found such a theorem. See section 1. We have also found theorems about a single algebraic set. See section 2. 5. ASCENDING CHAINS OF IDEALS. We take an ideal (in a commutative ring with unit) to be any subgroup which is closed under multiplication by any ring element. Thus the smallest ideal is {0}. The ideal {0} is generated by the empty set (and also by {0}). The degree of an ideal in a polynomial ring is the least d such that the ideal has a set of generators all of which have degree at most d. Here is Seidenberg's theorem on ascending chains: THEOREM 5.1. For all k,p >= 1 there exists n >= 1 such that the following holds. For any field F, there is no strictly ascending sequence of ideals in F[x_1,...,x_k] of length n, where the i-th ideal has degree at most p+i (i.e., a set of generators each of which has degree at most p+i). Actually, Seidenberg states this for any bound on the degree of the i-th ideal. Exactly analogous results hold. We prefer to state this as above because of our interest in lower bounds and the elementary nature of the result. Seidenberg established a primitive recursive bound only for k <= 2, and opently doubted whether a primitive recursive bound exists for even k = 3. He established no lower bounds. We first wish to reduce this to a more manageable form. There is a reasonable function G(k,i) such that for any field F, every ideal in F[x_1,...,x_k] of degree at most i has a set of generators, each of degree at most i, where there are at most G(k,i) generators. This function is obtained by the obvious linear algebra. LEMMA 5.2. For all k,p >= 1 there exists n >= 1 such that the following holds. Let F be a field. P[i,j] be a doubly indexed set of polynomials in F[x_1,...,x_k], where i,j satisfy 1 <= i <= n and 1 <= j <= G(k,p+i). Assume P[i,j] has degree at most p+i. Then there exists 1 <= i <= n such that for all 1 <= j <= G(k,p+i), P[i,j] lies in the ideal generated by the P[i',j'] where i' < i. Furthermore, there exists 1 <= i <= n such that for all 1 <= j <= G(k,p+i), P[i,j] can be written as an ideal element from the P[i',j'], i' < i, using polynomial coefficients of degrees at most n. It is interesting to also consider THEOREM 5.3. For all k,p >= 1 there exists n >= 1 such that the following holds. For any field F, there is no strictly ascending sequence of ideals in F[x_1,...,x_k] of length n, where the i-th ideal has degree p+i. Upper bounds for Theorem 5.2 give upper bounds for Theorem 5.3 and for Theorem 5.1. The lower bound for Theorem 5.3 also follows immediately from those given in section 1. We now give primitive recursive bounds for each k, in Theorem 5.2, using more logic. Consider the Pi-0-2 sentence *for all k,p >= 1 there exists n >= 1 such that T[k,p,n] is inconsistent.* Fix k. For any p, the least size of an inconsistency in T[k,p,n] is clearly an upper bound on the relevant number in Theorem 1.1, because already n is such an upper bound. But that least size is a primitive recursive function of p because the above statement is provable in WKL_0 for any given k. This is because the compactness theorem, the completeness theorem, and the Hilbert basis theorem are all provable in WKL_0 {In the manuscript with proofs, I formulate an appropriate first order theories T[k,p,n]}. 6. CHAINS OF ALGEBRAIC SETS - PRIMITIVE RECURSIVE BOUNDS. Let F be a field and k >= 1. An algebraic subset of F^k is the set of simultaneous zeros of some finite set of polynomials in k variables over F. The degree of an algebraic set is taken here to mean the least d such that it is the set of simultaneous zeros of some finite set of polynomials in k variables of degree at most d. THEOREM 6.1. For all k,p >= 1 there exists n >= 1 such that the following holds. For any field F, there is no strictly decreasing sequence of algebraic sets in F^k of length n, where the i-th algebraic set has degree at most p+i (i.e., is the zero set of a finite system of k variable polynomials over F of degree at most p+i). Theorem 6.1 follows immediately from Theorem 5.1, and our upper bounds for Theorem 5.1 also serve as upper bounds for Theorem 6.1. It is interesting to also consider THEOREM 6.2. For all k,p >= 1 there exists n >= 1 such that the following holds. For any field F, there is no strictly ascending sequence of ideals in F[x_1,...,x_k] of length n, where the i-th algebraic set has degree p+i. Upper bounds for Theorem 6.2 obviously give upper bounds for Theorem 5.3. Lower bounds for Theorem 6.2 follow immediately from those given in section 3. ********** This is the 71st in a series of self contained postings to FOM covering a wide range of topics in f.o.m. Previous ones are: 1:Foundational Completeness 11/3/97, 10:13AM, 10:26AM. 2:Axioms 11/6/97. 3:Simplicity 11/14/97 10:10AM. 4:Simplicity 11/14/97 4:25PM 5:Constructions 11/15/97 5:24PM 6:Undefinability/Nonstandard Models 11/16/97 12:04AM 7.Undefinability/Nonstandard Models 11/17/97 12:31AM 8.Schemes 11/17/97 12:30AM 9:Nonstandard Arithmetic 11/18/97 11:53AM 10:Pathology 12/8/97 12:37AM 11:F.O.M. & Math Logic 12/14/97 5:47AM 12:Finite trees/large cardinals 3/11/98 11:36AM 13:Min recursion/Provably recursive functions 3/20/98 4:45AM 14:New characterizations of the provable ordinals 4/8/98 2:09AM 14':Errata 4/8/98 9:48AM 15:Structural Independence results and provable ordinals 4/16/98 10:53PM 16:Logical Equations, etc. 4/17/98 1:25PM 16':Errata 4/28/98 10:28AM 17:Very Strong Borel statements 4/26/98 8:06PM 18:Binary Functions and Large Cardinals 4/30/98 12:03PM 19:Long Sequences 7/31/98 9:42AM 20:Proof Theoretic Degrees 8/2/98 9:37PM 21:Long Sequences/Update 10/13/98 3:18AM 22:Finite Trees/Impredicativity 10/20/98 10:13AM 23:Q-Systems and Proof Theoretic Ordinals 11/6/98 3:01AM 24:Predicatively Unfeasible Integers 11/10/98 10:44PM 25:Long Walks 11/16/98 7:05AM 26:Optimized functions/Large Cardinals 1/13/99 12:53PM 27:Finite Trees/Impredicativity:Sketches 1/13/99 12:54PM 28:Optimized Functions/Large Cardinals:more 1/27/99 4:37AM 28':Restatement 1/28/99 5:49AM 29:Large Cardinals/where are we? I 2/22/99 6:11AM 30:Large Cardinals/where are we? II 2/23/99 6:15AM 31:First Free Sets/Large Cardinals 2/27/99 1:43AM 32:Greedy Constructions/Large Cardinals 3/2/99 11:21PM 33:A Variant 3/4/99 1:52PM 34:Walks in N^k 3/7/99 1:43PM 35:Special AE Sentences 3/18/99 4:56AM 35':Restatement 3/21/99 2:20PM 36:Adjacent Ramsey Theory 3/23/99 1:00AM 37:Adjacent Ramsey Theory/more 5:45AM 3/25/99 38:Existential Properties of Numerical Functions 3/26/99 2:21PM 39:Large Cardinals/synthesis 4/7/99 11:43AM 40:Enormous Integers in Algebraic Geometry 5/17/99 11:07AM 41:Strong Philosophical Indiscernibles 42:Mythical Trees 5/25/99 5:11PM 43:More Enormous Integers/AlgGeom 5/25/99 6:00PM 44:Indiscernible Primes 5/27/99 12:53 PM 45:Result #1/Program A 7/14/99 11:07AM 46:Tamism 7/14/99 11:25AM 47:Subalgebras/Reverse Math 7/14/99 11:36AM 48:Continuous Embeddings/Reverse Mathematics 7/15/99 12:24PM 49:Ulm Theory/Reverse Mathematics 7/17/99 3:21PM 50:Enormous Integers/Number Theory 7/17/99 11:39PN 51:Enormous Integers/Plane Geometry 7/18/99 3:16PM 52:Cardinals and Cones 7/18/99 3:33PM 53:Free Sets/Reverse Math 7/19/99 2:11PM 54:Recursion Theory/Dynamics 7/22/99 9:28PM 55:Term Rewriting/Proof Theory 8/27/99 3:00PM 56:Consistency of Algebra/Geometry 8/27/99 3:01PM 57:Fixpoints/Summation/Large Cardinals 9/10/99 3:47AM 57':Restatement 9/11/99 7:06AM 58:Program A/Conjectures 9/12/99 1:03AM 59:Restricted summation:Pi-0-1 sentences 9/17/99 10:41AM 60:Program A/Results 9/17/99 1:32PM 61:Finitist proofs of conservation 9/29/99 11:52AM 62:Approximate fixed points revisited 10/11/99 1:35AM 63:Disjoint Covers/Large Cardinals 10/11/99 1:36AM 64:Finite Posets/Large Cardinals 10/11/99 1:37AM 65:Simplicity of Axioms/Conjectures 10/19/99 9:54AM 66:PA/an approach 10/21/99 8:02PM 67:Nested Min Recursion/Large Cardinals 10/25/99 8:00AM 68:Bad to Worse/Conjectures 10/28/99 10:00PM 69:Baby Real Analysis 11/1/99 6:59AM 70:Efficient Formulas and Schemes 11/1/99 1:46PM

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