Subalgebra analogue of Standard bases for ideals in $K[[t_{1},t_{2},\ldots,t_{m}]][x_{1},x_{2},\ldots,x_{n}]$

Abstract: In this paper, we develop a theory for Standard bases of K-subalgebras in K[[t1, t2, . . . , tm]][x1, x2, ..., xn] over a field K with respect to a monomial ordering which is local on t variables and we call them Subalgebra Standard bases. We give an algorithm to compute subalgebra homogeneous normal form and an algorithm to compute weak subalgebra normal form which we use to develop an algorithm to construct Subalgebra Standard bases. Throughout this paper, we assume that subalgebras are finitely generated.


Introduction, notation and definition
For the study of the structure of ideals in a polynomial ring K[x 1 , x 2 , ..., x n ] over a field K, Bruno Buchberger presented a concept of Gröbner bases with respect to global monomial orderings (Indeterminates x i are greater than 1, ∀i) [8]. In [8], Buchberger gave an algorithm called Buchberger Algorithm For the computation of Gröbner bases, based on the multivariate division algorithm (Normal form algorithm). The concept of Gröbner bases played an important role in the field of computational algebraic geometry and computational commutative algebra. Moreover, this concept was introduced for polynomial ring over a noetherian integral domain [10]. This concept is extended for the localization of K[x 1 , x 2 , ..., x n ] in [1], and termed Standard bases. The idea of Standard bases is tied with local monomial orderings (where indeterminates x i are less than 1, ∀i). They modified the idea of Normal form algorithm with respect to local monomial ordering to ensure the termination. It is an ecart based algorithm (for details, see Chapter 1 of [1]) known as Mora's algorithm (weak normal form algorithm) [3]. Furthermore, the study was made for Standard bases of ideals in a formal power series ring K[[t 1 , t 2 , ..., t n ]] in [2] with respect to local monomial orderings. Later, a theory of Standard bases for ideals in a more general mixed ordered indeterminates ring K[[t 1 , t 2 , . . . , t m ]][x 1 , x 2 , ..., x n ] was introduced with respect to a monomial ordering local on t indeterminates [5]. The concept of Standard bases in K[x 1 , x 2 , ..., x n ] (K[[t 1 , t 2 , . . . , t m ]]) is a special case of Standard bases in K[[t 1 , t 2 , . . . , t m ]][x 1 , x 2 , ..., x n ] for m = 0 (n = 0).
Subsequent to the concept of Gröbner bases, a concept of bases for subalgebras in K[x 1 , x 2 , ..., x n ] was introduced by Robianno and Sweedler termed as Sagbi bases [4]. Similar to Gröbner bases, this concept is also tied with global monomial orderings. The algorithm for the construction of Sagbi bases, is based on Sagbi Normal form algorithm which is the subalgebra analogue of Normal form algorithm of ideals. The idea of Sagbi bases has been generalized in polynomial ring over a noetherian integral domain [7]. Later, the concept of Standard bases was introduced for complete subalgebras in the formal power series ring K[[x 1 , x 2 , ..., x n ]] with respect to local monomial orderings [2]. The theory of Sagbi bases is extended to the localization of K[x 1 , x 2 , ..., x n ], called Sasbi bases 1 in [6]. As with Standard bases, this idea is also tied with local monomial orderings. They also presented the subalgebra version of Mora's algorithm, termed as Sasbi Normal form, which is also an ecart based algorithm.
Similar to the case of Standard bases for ideals [5], it is natural to ask for a theory of Standard bases for subalgebras in K[[t 1 , t 2 , . . . , t m ]][x 1 , x 2 , ..., x n ]. In this paper, we present the subalgebra analogue of Standard bases for ideals in K[[t 1 , t 2 , . . . , t m ]]-[x 1 , x 2 , ..., x n ], termed as "Subalgebra Standard bases". Similar to the Standard bases, we develop the idea of these bases with respect to a monomial ordering local on t indeterminates. As with Sagbi bases, these bases could be infinite for finitely generated subalgebras (see Example 3.6). The concept of Sagbi bases (assume x i 1, 1 ≤ i ≤ n) and Sasbi bases (assume  [6]). The theory of Subalgebra Standard bases (tied with mixed orderings), which we have introduced in this paper, unifies the previous theories (tied with global and local orderings). This theory is more general as previous theories could be seen as its special cases. It could also be used to solve problems like sublagebra membership problem in a mixed ring K[[t 1 , t 2 , . . . , t m ]][x 1 , x 2 , ..., x n ].
The structure of this paper is as follows. In the start, we give basic notations and terminologies and introduce the concept of Subalgebra Standard bases (Definition 1.7). The idea of normal form is very important to characterize subalgebra bases algorithmically. For this purpose, in Section 2, first we present an algorithm (Algorithm 2.3) to compute subalgebra homogenenous normal form for xhomogeneous 2 inputs. Due to x-homogeneity, the sequence of terms (obtained after each reduction) would have same x-degree and it would be convergent with respect to < t >-adic topology. Based on this algorithm, we give a weak subalgebra normal form algorithm (Algorithm 2.5), which is one of the key ingredients for the construction of Subalgebra Standard Bases. The weak subalgebra normal form can be seen as a combination of Sagbi normal form and Sasbi normal form. For the termination of this algorithm, for input G ⊂ S [x 1 , x 2 , . . . , x n ] we assume that an x-homogeneous S -subalgebra S [G * ] admits a finite Sagbi basis, where S = K[[t 1 , t 2 , . . . , t m ]]. Then, finally in Section 3, we provide an algorithm to compute Subalgebra Standard bases with the support of algebraic relations between leading terms of elements of the given input.
polynomial ring in indeterminates x with coefficients in the formal power series ring K [[t]]. Moreover, we use the notation t α for t α 1 1 t α 2 2 ...t α m m and x β for x β 1 1 x β 2 2 ...x β n n with α = (α 1 , α 2 , . . . , α m ) ∈ N m and β = (β 1 , β 2 , . . . , β n ) ∈ N n . Definition 1.1. A monomial ordering on the set of monomials Mon of R is a total ordering on the same set such that for all α, α , α " ∈ N m and β, β , β " ∈ N n We say a monomial ordering t-local if its restriction to the set of monomials of K[[t]] is local.
We call a t-local monomial ordering a t-local weighted degree ordering if there is a weight vector w = (w 1 , w 2 , ..., w m+n ) ∈ R m ≤0 × R n such that for all α, α ∈ N m and β, β , ∈ N n , the scalar product appears as: Let be a t-local monomial ordering. A non-zero element f of R can be viewed as: Now, we define a K-subalgebra 3 of R and its leading subalgebra. Definition 1.3. Let be a t-local monomial ordering on R and a subset G ⊆ R, then A = K[G] is a subalgebra of R generated by G. Naturally, the elements of A could be viewed as polynomials in terms of elements of G with coefficients in K. We define the leading subalgebra of G generated by LM(G) = {lm(g) | g ∈ G} as: is called a finitely generated subalgebra. Throughout this paper, we work with finitely generated subalgebras.
.., g k } be a subset of R. For a = (a 1 , a 2 , . . . a k ) ∈ N k , we call a power product of g i s a G-monomial, i.e., G a = g a 1 1 g a 2 1 . . . g a k k Remark 1.6. Any element f of subalgebra K[G] could be viewed as a finite sum in terms of Now, we define a Subalgebra Standard basis for a subalgebra of R as given in Definition 1.7.
Definition 1.7. Let be a t-local monomial ordering and A be a subalgebra of R. A Standard basis of A is a subset G ⊆ A such that in(G) = in(A) i.e. for any f ∈ A, lm( f ) ∈ in(G).
Note that in(G) need not be equal to in(A), i.e., not every generating set of the subalgebra is a Subalgebra Standard basis as we can see from the following example.
We can see that lm( f ) = x 6 1 t 2 in(G) and hence G is not a Subalgebra Standard basis of A.
Later, for the weak subalgebra normal form algorithm, we need the concept of multiplicative set and ecart defined as follows: Let be a t-local monomial ordering and A be a subalgebra of R, then we define the multiplicative set for A as: with respect to a t-local monomial ordering. Now, we present the concept of homogenization and dehomogenization of elements of R in only x indeterminates with respect to another indeterminate x 0 .
with | β | +γ = d for every term of f * and we define dehomogenization of F ∈ R * as: Let be a t-local monomial ordering. We define its homogenization * which is also a t-local monomial ordering, as: Now, we present some results on the relationship between elements and their homogenization and dehomogenization. Lemma 1.12. Let f ∈ R and G ⊂ R and an x- Proof. The proof of Parts (a)-(c) would be similar to the proof holds for polynomials (see [9]). Part (d) can be obtained by replacing f by F * and f * by F in Part(c). For Part(e), first assume For the converse, let G be the set such that e = ecart( f )−ecart( i c i G a i ) ≥ 0. From our assumption lm( f ) = lm( i c i G a i ) and Part (c) above, we get our required result.
To present the theory of Subalgebra Standard bases, we need a subalgebra reduction process (discussed in section 2). Now, we list the conditions the subalgebra reduction with its normal form may satisfy.
The above representation satisfies the following conditions (with respect to ): Indeterminate conditions: Homogeneous eterminate condition: HDC the above sum of G-monomials and r are x-homogeneous of x-degree and equal to deg x ( f ) With any of the conditions above, we call r a subalgebra normal form of f and if this is zero, we call such representation a Subalgebra Standard representation. Definition 1.14. Let be a t-local monomial ordering and u ∈ S = { f ∈ R | lm( f ) = 1}. Then under any of the above conditions, we call a subalgebra normal form r of u · f a weak subalgebra normal form of f with respect to G ⊂ R.

Subalgebra reduction process
In this section, we discuss the reduction process in subalgebras of R to compute subalgebra normal form with respect to t-local monomial orderings. For an x-homogeneous element in R, first, we present a theorem that shows the existence of subalgebra homogeneous normal form with respect to set of x-homogeneous elements in R along with an algorithm for its computation. Second, we present an algorithm to compute weak subalgebra normal form of any element in R. This algorithm is a key ingredient for the computation of Subalgebra Standard bases in R. Proof. We set f 0 = f and for v > 0 we define recursively satisfies (DC1), (DC2) and (HDC). The above representation of f v−1 used in 2.1 exists since power products of lt(g i s) are involved in this representation. Now, we want to show that the sequences ( f v ) ∞ v=0 and (r v ) ∞ v=1 converge to zero in the < t >-adic topology. By Lemma(2.3) [5], there exists a t-local weighted degree ordering w with weight w ∈ Z <0 × Z n for which lm(g i ) = lm(g i ) w (leading monomials with respect to w ) for all i, so after replacing by w , we get the same sequences ( f v ) ∞ v=0 , (r v ) ∞ v=1 since only power products of lm(g i s) are involved in their construction. In particular, (2.2) will satisfy (DC1), (DC2) and (HDC) with respect to w . Due to (HDC), f v is again x-homogeneous of x-degree equal to f v−1 . Moreover, (DC1)+(DC2)⇒ (IC1), so for all i From Lemma 2.4 [5], ( f v ) ∞ v=0 converges to zero in the < t >-adic topology and hence by construction (r v ) ∞ v=1 also converges to zero. But then r := of G monomials. The terms of r − r cannot be further reduced since r and r satisfy (DC2). Therefore, the representation of r − r is only possible if r − r is zero, i.e, r = r.
and be a t-local lexicographical ordering with x 1 x 2 1 t 1 t 2 . Furthermore, let f = t 1 , g 1 = t 1 −t 2 and g 2 = t 1 −(t 1 ) 2 be the elements of R. We can see that every representation of f in terms of g i s is not the one we require. For example, f = g 2 + g 2 1 + (2t 1 t 2 − t 2 2 ) does not satisfy DC1. However, there is a unique representation f = g 1 + t 2 which satisfies every condition.
On the basis of Theorem 2.1, we now present an algorithm to compute subalgebra homogeneous normal form. Instructions: p∈T fv p such that p = c p lt(G a p ) for some a p ∈ Z k ≥0 ; r and be a t-local lexicographical ordering with x 1 x 2 1 t 1 t 2 . Also, let . . be the elements of R. Here f, g 1 , g 2 are x-homogeneous. Note that lt(g 1 ) = x 1 and lt(g 2 ) = t 1 . Table 1 shows the subalgebra reduction through Algorithm 2.3. Step The representation given by the algorithm is: For f ∈ R and G = {g 1 , g 2 , . . . , g k } ⊂ R, we now present an algorithm to compute weak subalgebra normal form of f with respect to G, which plays an important role for the characterization of Subalgebra Standard bases. We assume that A = S [G * ] (as an S -subalgebra of S [x * ]) admits a finite Sagbi basis with respect to * , where S = K[[t]] 4 . Algorithm 2.5. (WNF) Let be any t-local monomial ordering. Input: f ∈ R and G = {g 1 , g 2 , . . . , g k } ⊂ R Output: r ∈ R, a weak subalgebra normal form of f with respect to G. Instructions: Remark 2.6. Algorithm 2.5 works on the assumption that we are able to produce subalgebra homogeneous normal form as we can see that it relies on HNF algorithm. If G ⊂ R and f ∈ R, then after applying Algorithm 2.5, we can write as u · f = i c i G a i + r for some u ∈ S ,A and G-monomials G a i ; where r = WNF( f, G).
For the termination part of Algorithm 2.5, we first introduce a few notations.
, for which lt x * (g * )is a product of power series as a coefficient with greatest x * -power product with respect to * ordering. The leading subalgebra generated by LT Then we can write g as ( Remark 2.9. For f ∈ R and G ⊂ R, note that we have a compatibility between lm and lt x * in a sense that lt x * ( f * ) = lt x * (G a ) * implies lm( f ) = lm(G a ). Now we prove the termination and correctness of Algorithm 2.5. Proof. Termination: In order to see the termination of the algorithm, first, we need to show that stops. We use homogenization to prove it. By assumption, a Sagbi basis for A is finite implies that the ascending chain of initial sublagebras . . must terminate (see [4] for further details). If this chain terminates, then It shows that T * v itself becomes stable for v ≥ N and so the algorithm continues to run with the fixed T * . Since p * N ∈ D * N , therefore by Remark 2.9, p N ∈ D N , so the Chain(2.3) continues with the fixed T , i.e., T v = T N for all v ≥ N, where N ∈ Z ≥0 . Algorithm 2.3 ensures that lm(R N ) * K[T N ] which implies that D N+1 is empty and hence the algorithm terminates.
The above equation, inequality (2.4) and Eq (2.7) imply that the representation in Eq (2.9) satisfies (IC1) and (IC2). Therefore, r is the weak subalgebra normal form. Case e > 0, By Theorem 2.1, x where r is a weak subalgebra normal form of R * and u ∈ S ,K[T ] (lm(u ) = 1), satisfies (IC1) and (IC2) with T a j involves only g i s and T a s involves f as well. It implies for all j and s, lm(R * ) = lm(u · R * ) lm(T a j ) and lm(R * ) = lm(u · R * ) lm(T a s ).
Using inquality (2.14), we have lm( f ) lm(T a j ) and lm( f ) lm(T a s ).
We can see that there are no non-trivial algebraic relations (see [7] for details). Hence G * is a Sagbi basis which certifies the termination of Algorithm WNF. Here ecart( f ) = 2, ecart(g 1 ) = 4 and ecart(g 2 ) = 0. Table 2 shows the subalgebra reduction through Algorithm 2.5. Table 2. Example 2.10: Subalgebra Weak Reduction of f . Step So we get a weak subalgebra representation of f as:

Construction of Subalgebra Standard bases
For Subalgebra Standard bases criterion, we define a notion of algebraic relations for G ⊂ R. For this, we define an evaluation map π: K[Y] → K[LM(G)] via y i → lm(g i ); where the cardinality of Y = {y 1 , y 2 , · · · } is same as that of G.
Definition 3.1. Let G = {g 1 , g 2 , . . . , g k } ⊂ R. The set of algebraic relations of G denoted by AR(G) is the kernel of above map π, i.e., The strict inequality holds since P j ∈ AR(G). We may assume that w = w j , where 1 ≤ j ≤ m, therefore for each j, we have Let X j = max l j j=1 {lm( f j (G)lm(G a q j )} be the height of the right hand side in Eq (3.3), then using (3.2) we have X j ≺ max m j=1 {ht( f j (G))ht(P j (G))} = X. Now, the Eqs (3.1) and (3.3) imply that: We see that X j ≺ X ; for all 1 ≤ j ≤ m. Therefore, ht(sum 1 ) = max m j=1 X j ≺ X. By the choice of α, ht(sum 2 ) ≺ X, which is a contradiction to our assumption of a representation of g with smallest possible height. Thus, G is a Subalgebra Standard basis of K[G].