PAPER On the Degree of Multivariate Polynomials over Fields of Characteristic 2

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1 IEICE TRANS. INF. & SYST., VOL.E88 D, NO.1 JANUARY PAPER On the Degree of Multivariate Polynomials over Fields of Characteristic 2 Marcel CRASMARU a), Nonmember SUMMARY We show that a problem of deciding whether a formula for a multivariate polynomial of n variables over a finite field of characteristic 2 has degree n when reduced modulo a certain Boolean ideal belongs to P. When the formula is allowed to have succinct representations as sums of monomials, the problem becomes P-complete. key words: degree of multivariate polynomials, finite field of characteristic 2, parity P-complete, Hamilton path 1. Introduction The class of languages accepted by polynomial-time bounded unambiguous alternating Turing Machines, i.e., UAP, was introduced by Niedermeier and Rossmanith in [2]. Our object is to analyze the next decision problem: DEGREE-MOD-I 2 : INSTANCE: A formula for a polynomial f F[x 1,...,x n ], over a finite field F of characteristic 2. QUESTION: Doesf have degree n when reduced modulo the Boolean ideal I 2 = x 2 1 x 1,...,x 2 n x n? Using the results of [3], it was shown in [4] that this problem is polynomial-time many-one hard for UAP. The fact that UAP contains FewP (see [2]) makes the DEGREE-MOD-I 2 problem computationally hard. But what is the exact complexity of DEGREE-MOD-I 2? One may observe that f reduces to 1 or 0 iff it is a tautology or it is unsatisfiable. It follows that the related question of whether thedegreemodi 2 is positive is NP-hard. However, it does not follow simply that our problem of whether the degree is n is NP-hard. The problem of deciding whether a given set of polynomials reduces to 1 under the Gröbner basis algorithm belongs to the second level of the polynomial hierarchy [5]. It is easy to check that {x 2 1 x 1,...,x 2 n x n } is a Gröbner basis for I 2, but showing that DEGREE-MOD-I 2 belongs to the second level of the polynomial hierarchy would put UAP inside PH, which is also questionable. In the following, we show that that DEGREE-MOD-I 2 is decidable in P. Moreover, for a slightly general class of formulas, i.e., formulas allowing succinct representations of sums of monomials, we show that the corresponding DEGREE-MOD-I 2 problem is P- complete. We also introduce a generalized problem, called Non-Reducible-Monomial, and prove that it is P- Manuscript received March 18, Manuscript revised August 6, The author is with the Dept. of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo, Japan. a) marcel@is.titech.ac.jp complete. 2. Preliminaries Let F be a finite field of characteristic 2 and let R = F[x 1,...,x n ] be the ring of polynomials with coefficients in F and variables x 1,...,x n. Without losing generality, we may regard F = {0, 1}, and addition and multiplication in F are the mod 2 addition and multiplication in {0, 1}. We assume some canonical code for polynomials, which is essentially syntactic representation of polynomials encoded by binary strings in some reasonable way. For simplifying our discussion, we assume that monomials are written as x 1 x 1 x 3 x 3 x 3 instead of x 2 1 x3 3, though we often use the latter notation in our discussion. This representation is not succinct, and some code may become exponentially long; but this inefficiency of encoding doesnotaffect the discussion of this paper. Except for expanding exponents, no extra algebraic computation is made for obtaining our canonical code; hence, for example, x 1 x 3 x 1 x 2 and x 1 x 1 x 2 x 3 are encoded differently. Clearly, two monomials with different canonical codes are the same if one transform to another by permuting variables. Given an element f of R we denote by f the length of the binary encoding (representation) of f. Let M(R, I 2 ) be the set of monomials of the factor ring R/I 2.Asanexample,M(R, I 2 ) = {0, 1, x 1, x 2, x 1 x 2 }, when R = F[x 1, x 2 ], and I 2 = x 2 1 x 1, x 2 2 x 2. Let R = F[x 1,...,x n ]. We would like to generalize expressions for formulas by introducing Σ-expressions. First, we give a syntactic definition. A Σ-formula is defined inductively as follows. Definition 2.1: 1. Any monomial p R is a Σ-formula. 2. If S is a finite set of canonical codes of monomials in R,then p S p is a Σ-formula. 3. If φ and ψ are Σ-formulas, then φ + ψ is a Σ-formula. 4. If φ and ψ are Σ-formulas, then φ ψ is a Σ-formula. For a set S of canonical codes of monomials in R, the meaning of p S p is simply the sum of all (canonical codes of) monomials appearing in S. For example, for S = {x 1 x 2 x 3, x 2 x 1 x 3, x 3 x 2 x 1 }, p S p is equivalent to x 1 x 2 x 3 + x 2 x 1 x 3 + x 3 x 2 x 1. Notice here that a semantically same monomial may appear more than once as different canonical codes in S ; in the above example, p S p is essentially adding the same monomial x 1 x 2 x 3 three times. The number of such duplicates in one set is bounded by the numberofdifferent canonical codes; for example, while x 3 1 Copyright c 2005 The Institute of Electronics, Information and Communication Engineers

2 104 IEICE TRANS. INF. & SYST., VOL.E88 D, NO.1 JANUARY 2005 (= x 1 x 1 x 1 ) appears at most once, x 1 x 2 x n can appear at most n! times. For specifying a Σ-formula such as f = (Σ m S 1 m + x 1 x 5 )(Σ m S 2 m+x 2 3 x 7x 8 +Σ m S 3 m), we need to specify S 1, S 2, and S 3. In this paper, we specify a set S of canonical codes of monomials in R by giving a Boolean circuit C S.Loosely speaking, C S decides whether a given binary code for some canonical codeof monomial is in S. More precisely, we use slightly simpler way to encode canonical codes of monomials. Since S is a finite set, we can define l to be the number of variables appearing in the longest monomial in S.Also assume that n, the number of variables, is given separately as a part of the description of the whole formula. Then we can encode each canonical code of monomial in S by using a l log(n + 1) bit binary string. For example, in the case that n = 3andl = 4 (for a given S in a given formula f ), a monomial (i.e., its canonical code) x 2 x 1 x 3 in S is expressed by , where the last 00 is for padding. Similarly, x 3 x 2 x 1 x 2 by , x 1 x 3 by , and so on. Then the circuit C S for specifying S is a circuit that takes l log(n+1) bits as an input and determines whether the corresponding canonical code of monomial is in S.Forexample, if S = {x 1 x 3, x 2 x 1 x 3, x 3 x 2 x 1 x 2, x 3 x 1 } (and n = 3), then C S says 1 (i.e., yes ) to , , , , and 0 (i,e., no ) to the other inputs. Clearly, some S may need very large circuit C S for specifying it. On the other hand, even if S contains exponentially many elements (w.r.t. n), it may be still possible that the size of C S is not so large, say, within some polynomial in n. What is important here is that when C S is given, one can nondeterministically guess every canonical code in S within polynomial-time in the size of C S. Since S may contain exponentially many elements, the nondeterminism is needed in general. Finally we formally define the way to give a Σ-formula as an input. Any Σ-formula f is represented by a binary string encoding a tuple ( f, C S 1,...,C S k ), where f is an expression of f and C S 1,...,C S k are circuits specifying the set of canonical codes of monomials used in the expression f. Thus, the input size of f is the length of the binary string encoding ( f, C S 1,...,C S k ). We are ready to define a generalized version of DEGREE-MOD-I 2 problem. Σ-DEGREE-MOD-I 2 : INSTANCE: AΣ-formula for a polynomial f F[x 1,...,x n ], over a finite field F of characteristic 2. QUESTION: Doesf have degree n when reduced modulo the Boolean ideal I 2 = x 2 1 x 1,...,x 2 n x n? We also consider another generalization of DEGREE-MOD-I 2, which is defined as follows for any ideal I. (It is easy also to see that DEGREE-MOD-I 2 is polynomialtime reducible to Σ-NON-RED-MONOMIAL-I 2.) Σ-NON-RED-MONOMIAL-I: INSTANCE: AΣ-formula f for a polynomial and a monomial m in R = F[x 1,...,x n ]. Also the ideal I is given as an oracle. QUESTION: Does f contain the monomial m when reduced modulo the ideal I? Here by an oracle for I, we mean an oracle set EQ(I)that contains all and only all pairs of monomials (m, m ) R 2 for which m m (mod I) holds. (An ideal I depends on R = F[x 1,...,x n ], in particular, the set of variables. Thus, precisely speaking, the problem Σ-NON-RED-MONOMIAL-I is not defined by a single ideal I but by a family of ideals, and the oracle should be given, at least, the information on the number of variables. For the simplicity, however, we discuss below by regarding I as a single ideal.) An oracle for the ideal is necessary in general because there are ideals I such that deciding whether two monomials are equivalent modulo I is infeasible. More precisely, deciding whether two monomials m and m are equivalent modulo I is computationally equivalent to deciding whether the binomial m m is in the ideal. The ideal membership problem remainsexpspace-hard when restricted to binomials [6]. Note here that we are not concerned in such hard cases; we consider only easy ideals I for which EQ(I) is decidable in polynomial-time, i.e., EQ(I) P. Our goal is to show that Σ-NON-RED-MONOMIAL-I is still as hard as P even in such an easy situation. In fact, the equivalence of monomials is trivial for all explicitly defined ideals occuring in the sequel. For example, we have that EQ(I 2 ) P. The class P EQ(I) is the class of languages L such that there exists a nondeterministic oracle machine M EQ(I) such that x L iff the number of accepting paths of the machine M EQ(I) on the input x is odd. In general, an oracle machine M A is defined as a multi-tape Turing machine with an input tape, an output tape, work tapes, and a query tape. The machine M A has three distinguished states: QUERY, YES and NO. During the computation, the oracle machine may enter the state QUERY, and depending of the membership of the string currently written on the query tape in the fixed oracle set A, the machine goes to the state YES, or to the state NO. In the next section we present a P EQ(I) algorithm that decides whether a Σ-formula contains a certain non-reducible monomial. 3. A P EQ(I) Algorithm Solving Σ-NON-RED- MONOMIAL-I Let I be an ideal in R = F[x 1,...,x n ] for which we assume an oracle set EQ(I) as stated in Σ-NON-RED-MONOMIAL-I problem. Let us consider the following nondeterministic algorithm that queries the oracle EQ(I), takes as input a Σ- formula f in R and two monomials m, m R, and outputs accept or reject. (Below we use p, m, m, m to denote monomials; a 1, a 2,andb to denote subformulas.) procedure findmonomial(f, m, m ) { 0 if( f = 0) {reject; halt;} 1 if( f = 1) { 2 if((m, m ) EQ(I)) { accept; halt;} 3 else { reject; halt;}

3 CRASMARU: ON THE DEGREE OF MULTIVARIATE POLYNOMIALS OVER FIELDS OF CHARACTERISTIC } 5 if( f = m and m is Monomial ) { 6 call findmonomial(1, m, m m ); 7 } 8 if( f = p S p) { 9 nondeterministically guess p S ; 10 call findmonomial(p, m, m ); 11 } 12 if( f = a 1 + a 2 ) { 13 nondeterministically guess i {1, 2}; 14 call findmonomial(a i, m, m ); 15 } 16 if( f = m b) { 17 call findmonomial(b, m, m m ); 18 } 19 if( f = ( p S p) b) { 20 nondeterministically guess p S ; 21 call findmonomial(p b, m, m ); 22 } 23 if( f = (a 1 + a 2 ) b) { 24 nondeterministically guess a i, i {1, 2}; 25 call findmonomial(a i b, m, m ); 26 } } // end procedure Here all if-conditions are about f s syntactic form; for example, the condition f = p S p of line 8 asks whether f is syntactically of this form. Hence, it is easy to decide these conditions. Here m and b denote a nontrivial monomial and a nontrivial formula respectively. Then it is clear that every formula f satisfies exactly one of these if-conditions. We can choose monomials and/or subformulas so that arguments g, a, andb in subsequent recursive calls findmonomial(g, a, b) at lines 6, 13, 17, 21, and 25 are uniquely determined. Furthermore, we can choose them so that g < f always holds; that is, g should be syntactically simpler than f. From this choice, it is clear that the algorithm reaches to either line 0 or 1 in f steps, when counting a query to the oracle EQ(I) as one step. By further observing that max( a, b ) f + m + m we conclude that findmonomial is an oracle nondeterministic polynomial-time algorithm with respect to the length of its input. Note that the oracle EQ(I) is queried at line 2. Let f be the formula obtained from f using the distributive law of the ring R, but without cancelling identical terms. For example, if f = (x 1 + x 2 )(x 1 + x 2 ), then f = x x 1 x 2 + x 1 x 2 + x 2 2. Similarly, any sum of the form p S p in f is expanded (only syntactically) in f. Lemma 3.1: For any given ideal I, a formula f, and monomials m and q, consider the execution of the algorithm findmonomial( f, m, q). The number of accepting paths in the execution is equal to the number of monomials m of f that satisfies the condition m m q (mod I). Proof: We fix a given ideal I. Let us denote by #M( f, m, q) the number of accepting paths in the execution of findmonomial( f, m, q). Denoteby#m( f, m, q) the number of monomials m of f verifying the condition m m q (mod I). We prove the lemma, that is, #M( f, m, q) = #m( f, m, q) for all monomials m and q, by structural induction on f.following Definition 2.1, first consider the base case, i.e., the case that f is a monomial. Then as one may see from lines 5 and 6, and line 2 of the algorithm, we reach the accepting state if and only if (m, f q) EQ(I) m f q (mod I); hence #M( f, m, q) = #m( f, m, q). The second case is the case that f = p S p, forsome set S of canonical codes of monomials. For all monomials p S, we may assume (by the induction hypothesis) that findmonomial(p, m, q) returns accept if and only if m p q (mod I). Lines 9 and 10 of the algorithm show us that the number of accepting paths of findmonomial( f, m, q) is precisely the number of monomials p S (i.e., p appearing in f ) satisfying the condition m p q (mod I). That is, #M( f, m, q) = #m( f, m, q). Third, consider the case that f = ψ + φ, #M(ψ, m, q) = #m( ψ, m, q) andthat#m(φ, m, q) = #m( φ, m, q). Then lines 13 and 14 of the algorithm shows us that #M( f, m, q) = #M(ψ, m, q) + #M(φ, m, q). On the other hand, from the way to define f,wehave#m( f, m, q) = #m( ψ, m, q)+#m( φ, m, q). Thus, we can conclude that #M(ψ, m, q) = #m( f, m, q). Finally, consider the case that f = ψ φ. Here we consider subcases depending on the form of the subformula ψ. (We may assume that ψ is not of the form ψ φ by considering the shortest factor of f as ψ.) Suppose first that f = p ψ, where p is a monomial. From the algorithm it follows that #M( f, m, q) = #M(ψ, m, p q). Then we have #M( f, m, q) = #m( ψ, m, p q), because we may assume (by the induction hypothesis) that #M(ψ, m, q ) = #m( ψ, m, q ) for all monomials m, q R. Observing that #m(p ψ, m, q) = #m( ψ, m, p q) andthat f = p ψ, wehave#m( ψ, m, p q) = #m( f, m, q). Hence we conclude that #M( f, m, q) = #m( f, m, q). Suppose next that f = ( p S p) ψ, where p is a monomial. Then the lines 20 and 21 of the algorithm show us that #M( f, m, q) = p S #M(p ψ, m, q). Following the same argument as above, we have that #M( f, m, q) = p S #M(p ψ, m, q) = p S #m( p ψ, m, q). On the other hand, we have #m( f, m, q) = p S #m( p ψ, m, q); thus #M( f, m, q) = #m( f, m, q) follows. The final case, i.e., the case that f = (ψ + π ) π can be treated in the same way as above, and is omitted. Theorem 3.2: Σ-NON-RED-MONOMIAL-I belongs to P EQ(I). Proof: Let M EQ(I) be an oracle nondeterministic TM that queries the oracle EQ(I), takes as an input a Σ-formula f and a monomial m, and computes findmonomial( f, m, 1). Such a machine can be easily designed using our algorithm. As a P EQ(I) machine, we consider that an input ( f, m) is accepted if and only if M EQ(I) has odd number of accepting

4 106 IEICE TRANS. INF. & SYST., VOL.E88 D, NO.1 JANUARY 2005 paths on this input. Consider the execution of this M EQ(I) on the given inputs f and m. The number of its accepting paths is #M( f, m, 1), which is by Lemma 3.1 #m( f, m, 1), that is, the number of monomials m of f such that m m 1 (mod I) holds. On the other hand, since the field F is of characteristic 2, the formula f contains the monomial m when reduced modulo the ideal I if and only if #m( f, m, 1) 0 (mod 2), or equivalently, #m( f, m, 1) 1 (mod 2). This is precisely the condition that M EQ(I) accepts ( f, m) asa P EQ(I) machine. Since the equivalence of monomials under I 2 is quite easy to decide, i.e., EQ(I 2 ) P, the following corollary is immediate. Corollary 3.3: Σ-DEGREE-MOD-I 2, DEGREE-MOD-I 2,and Σ-NON-RED-MONOMIAL-I 2 belong to P. 4. Σ-DEGREE-MOD-I 2 is P-Complete It is well known that the fully HAMILTON PATH problem is P-complete (for a proof see [1], page 442). INSTANCE: A digraph G = (V, E). QUESTION: Is the number of Hamilton paths of G odd? We show that HAMILTON PATH reduces polynomial-time many to one to Σ-DEGREE-MOD-I 2. Let G = (V, E) be a graph having n vertices and edges E V 2. We continue to assume that F is a finite field of characteristic 2, and let R = F[x 1,...,x n ] be the ring of polynomials with coefficients in F and variables x 1,...,x n. Let M(F, I 2 ) be the set of monomials of the ring R/I 2, where I 2 is the ideal x 2 1 x 1,...,x 2 n x n. We label vertices of G such that V = {x 1, x 2,...,x n } and consequently identify the edges of E correspondingly. To define a Σ-formula of R, we consider the following set S G of canonical codes of monomials: S G = {v 1 v 2 v n (v i, v i+1 ) E for all 1 i n 1}. Recall that, in the description of the Σ-formula, S G needs to be specified as a circuit C S G recognizing it. Such a circuit C S G is defined as follows. Note that all canonical codes in S G consists of n variables from the variable set {x 1,...,x n }. Hence, each one is encoded by a string of length n log(n + 1). A circuit C S G, given a binary string of length n log(n + 1) to its input gates, checks whether encoded v 1,...,v n satisfies the condition of S G ;thatis,it checks whether (v i, v i+1 ) E, foralli, 1 i n 1. It outputs yes if the condition is satisfied, and outputs no otherwise. It is easy to see that such a circuit can be constructed in polynomial-time from G. Now define f G by f G = m S G m. Then f G is a Σ-formula. Furthermore its representation ( f, C S G ) is computable from G within polynomial-time in its size G. Our reduction from HAMILTON PATH to Σ-DEGREE-MOD-I 2 is to transform G to ( f, C S G ). The next lemma shows that this indeed is a reduction from HAMILTONPATHtoΣ-DEGREE-MOD-I 2. Lemma 4.1: The number of Hamilton paths of G = (V, E) is odd if and only if f G contains the monomial x 1 x 2 x n when reduced modulo I 2. Proof: For a given G = (V, E), define T G by: T G = {m S G m x 1 x 2 x n (mod I 2 )}. That is, T G is the set of canonical codes of monomials m in S G that satisfies m x 1 x 2 x n (mod I 2 ). For the proof it suffices to show that there is a one-toone correspondence between the set of Hamilton paths of G and the elements of T G. Let v 1, v 2,...,v n be a Hamilton path of G. By definition, we have that (v i, v i+1 ) E for all 1 i n 1, and {v 1, v 2,...,v n } = {x 1, x 2,...,x n }. Hence the corresponding monomial code v 1 v 2 v n belongs to T G. It is also easy to see that any element of T G, i.e., a code x i1 x i2 x in S G representing the monomial x 1 x 2 x n, corresponds uniquely to some path of G. From Corollary 3.3 we deduce the following. Theorem 4.2: Σ-DEGREE-MOD-I 2 is P-complete. A straightforward corollary follows from Theorem 3.2. Corollary 4.3: Σ-NON-RED-MONOMIAL-Iis P-complete for any ideal I satisfying EQ(I) P. 5. Σ-NON-RED-MONOMIAL-I Remains P-Hard When Restricted to Normal Formulas One may observe that our proof that Σ-NON-RED-MONOMIAL-I is P-hard uses the fact that a Σ-formula may contain succinct representations of (possibly) exponential large sums of monomials. Here we show that one can restrict Σ-NON-RED-MONOMIAL-I to normal formulas, i.e., formulas with no Σ-expressions, but Σ-NON- RED-MONOMIAL-Istill remains P-complete for some ideals I satisfying EQ(I) P. In the sequel we consider only normal formulas, that is, a formula is given by modifying Definition 2.1 such that Σ-sums over sets of canonical codes of monomials (i.e., expressions of type 2 of Definition 2.1) are not allowed. Let NON-RED-MONOMIAL-I be the decision problem Σ-NON- RED-MONOMIAL-I restricted to normal formulas. It is clear that the algorithm stated in Sect. 3 works for the set of normal formulas, which proves the following fact. Lemma 5.1: NON-RED-MONOMIAL-I belongs to P EQ(I). Theorem 5.2: For any ideal I such that EQ(I) P, the problem NON-RED-MONOMIAL-I is P-complete.

5 CRASMARU: ON THE DEGREE OF MULTIVARIATE POLYNOMIALS OVER FIELDS OF CHARACTERISTIC Proof: We reduce HAMILTON PATH to NON-RED- MONOMIAL-I nt, where I nt is some simple ideal defined below. More specifically, we reduce a given graph G for HAMILTON PATH to an instance ( f G, x 1 x 2 x n )of NON-RED-MONOMIAL-I nt. For any given G = (V, E), we define f G as follows. For each vertex v of G, let in(v) andout(v) denotetheset of incoming and respectively outgoing edges of v. Without lose of generality we may assume that there are precisely two vertices a, b V such that out(a) = {a} {V \{a, b}}, in(b) = {V {a, b}} {b}, andin(a) = out(b) =. Ifsuch vertices fail to exist, we just add them to V and also add the necessary edges to E. It is easy to see that the number of Hamilton paths is invariant to the above transformation. Note also that any Hamilton path of such a graph always starts with the vertex a and ends with the vertex b. Let V = n and E = t. LetF be a finite field of characteristic 2, and let I nt = y 1 z 1 1,...,y t z t 1 be an ideal in the ring R = F[x 1,...,x n, y 1,...,y t, z 1,...,z t ]. It is easy to check that EQ(I nt ) P. We may assume that the vertices in V are labeled by x 1,...,x n such that a is labeled by x 1 and b is labeled by x n. On the other hand, we label the set E of edges by a subset of {y 1,...,y t } {z 1,...,z t } such that each edge e i = (x i j, x ik ) E is labeled by (y i, z i ), where 1 i t. In what follows we identify the vertices and the edges by their labels. Consider the following formula. f G = ( ) ( ) x 1 y j1 z i2 x 2 y j2 (y j1,z j1 ) out(x 1 ) [(y i2,z i2 ),(y j2,z j2 )] in(x 2 ) out(x 2 ) ( ) z ik x k y jk ( [(y ik,z ik ),(y jk,z jk )] in(x k ) out(x k ) (y in,z in ) in(x n ) z in x n ) Though Σ-sums are used in the above expression, f G should be expressed as a normal formula. Note that for any vertex v G one may construct the sets in(v), out(v), and in(v) out(v) within polynomial-time in G, the size of the graphg. Thus every Σ-sum of monomials involved in the expression of f G can be transformed to a normal one within polynomialtime in G. It follows that one can construct the formula f G within polynomial-time in G. As before, let f G denote the formula obtained from f G using the distributive law of the ring R, but without cancelling identical terms. Then we can naturally correspond each monomial in f G to some unique canonical code. We let S G be the set of such canonical codes of monomials appearing in f G, and define the following set T G = {m S G m x 1 x 2 x n (mod I nt )}. To finish the proof it suffices to show that there is a one-toone correspondence between the set of Hamilton paths of G and the elements of T G. Let p = x i1, x i2,...,x in be a Hamilton path of G (denoted by vertices), which is expressed by edges as (y j1, z j1 )(y j2, z j2 )...(y jn 1, z jn 1 ). Note that i 1 = 1andi n = n because any Hamilton path starts and ends with respectively vertex a and vertex b. For this Hamilton path p, consider the monomial m p = x i1 y j1 z j1 x i2 y j2 z jn 1 x n. Then it appears in f G ; thus the corresponding code belongs to S G. Also it is easy to see that x i1 y j1 z j1 x i2 y j2 z jn 1 x n x 1 x 2 x n (mod I nt ); hence we have m p T G. Moreover m p is uniquely determined by p; thatis,ifp p are two different Hamilton paths of G then, as canonical codes, m p and m p are different. It follows that every Hamilton path of G uniquely corresponds to some element of T G. Now it remains to show that any element of T G corresponds to a path of G. Let m be an element of T G. By looking at the expression of f G we can conclude that m must be of the form x 1 y j1 z i2 x 2 y j2 z in 1 x n 1 y jn 1 z in x n. Then from the definition of T G it follows that x 1 y j1 z i2 x 2 y j2 z in 1 x n 1 y jn 1 z in x n x 1 x 2 x n (mod I). But the only way to reduce the variables (y i j ) 1 j n 1 and (z i j ) 1 j n 1 modulo the ideal I nt is by grouping them in pairs corresponding to edges in the graph G. That is, m may be re-written as m = x j1 y k1 z k1 x j2 y k2 z k2 y kn 1 z kn 1 x jn such that (y ki, z ki ) is the label of the edge from x ji to x ji+1 for all 1 i n 1. It is now clear that to any element of T G corresponds a Hamilton path of G. This shows that the number of Hamilton paths of G is odd iff T G is odd iff f G contains the monomial x 1 x 2 x n when reduced modulo I nt and completes the proof of the theorem. 6. Complexity of DEGREE-MOD-I 2 and Conclusions As shown in Corollary 3.3, DEGREE-MOD-I 2 is decidable in P. We know that DEGREE-MOD-I2 is UAP-hard. Unfortunately, we can not prove that DEGREE-MOD-I 2 is complete for UAP or P. What we can prove, however, is that the problem that corresponds to DEGREE-MOD-I 2 for Σ- formulas, i.e., Σ-DEGREE-MOD-I 2,is P-complete. It still remains an open issue to precisely locate DEGREE-MOD-I 2 problem between UAP and P.Abetter result would be to prove that DEGREE-MOD-I 2 is UAPcomplete, by contrast to our results presented here. However, we have fully settled the complexity of a related problem; Σ-NON-RED-MONOMIAL-I is P-complete for any ideals for which the monomial equivalence problem is polynomial-time decidable. Acknowledgment I would like to thank Prof. Osamu Watanabe for his comments and suggestions. Also comments from referees were helpful for improving the presentation of the paper. References [1] C.H. Papadimitriou, Computational complexity, Addison-Wesley, [2] R. Niedermeyer and R. Rossmanith, Unambiguous computations and

6 108 IEICE TRANS. INF. & SYST., VOL.E88 D, NO.1 JANUARY 2005 locally definable acceptance types, Theor. Comput. Sci., vol.194, no.1-2, pp , [3] S. Aida, M. Crasmaru, K. Regan, and O. Watanabe, Games with a uniqueness property, STACS2002, Lecture Notes in Computer Science, vol.2285, pp , [4] M. Crasmaru, C. Glaßer, K. Regan, and S. Sengupta, On global unique alternations, Kenneth W. Regan Selected publications, [5] P. Koiran, Hilbert s Nullstellensatz is in the polynomial ierarchy, J. Complexity, vol.12, no.4, pp , [6] E. Mayr and A. Meyer, The complexity of the word problems for commutative semigroups and polynomial ideals, Adv. Math., vol.46, pp , Marcel Crasmaru received in 2001 Master of Science from Tokyo Institute of Technology, Dept. of Information Sciences, and received Dr. of Science from Tokyo Institute of Technology, Dept. of Information Sciences in He is currently with NTT Communications. He is interested in computational complexity theory and theoretical aspects of computer games.