Research article

On the Tame automorphisms of differential polynomial algebras

  • Let R{x,y} be the differential polynomial algebra in two differential indeterminates x,y over a differential domain R with a derivation operator δ. In this paper, we study on automorphisms of the differential polynomial algebra R{x,y} with one derivation operator. Using a method in group theory, we prove that the Tame subgroup of automorphism of R{x,y} is the amalgamated free product of the Triangular and the Affine subgroups over their intersection.

    Citation: Zehra Velioǧlu, Mukaddes Balçik. On the Tame automorphisms of differential polynomial algebras[J]. AIMS Mathematics, 2020, 5(4): 3547-3555. doi: 10.3934/math.2020230

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  • Let R{x,y} be the differential polynomial algebra in two differential indeterminates x,y over a differential domain R with a derivation operator δ. In this paper, we study on automorphisms of the differential polynomial algebra R{x,y} with one derivation operator. Using a method in group theory, we prove that the Tame subgroup of automorphism of R{x,y} is the amalgamated free product of the Triangular and the Affine subgroups over their intersection.


    For a prime p1(mod3), let Fp be the finite field of residues (modp), let G be the multiplicative group of non-zero residues (modp) and let H be the subgroup of non-zero cubic residues (modp). For any aG, we defined the sums

    S(a)=p1k=0e(ak3/p)

    and

    G(χ)=p1k=1χ(k)e(k/p),

    where χ is a multiplicative character of order 3 over Fp and e(x)=e2πix in this paper. Both S(a) and G(χ) are called Gauss sums of order 3. Gauss sums is very important in the analytic number theory and related research filed. Many scholars studied its properties and obtained a series of interesting results (see [5,6,8,9,10,11,13]).

    Let zGH. By a classical result of Gauss [4] (also see Theorem 4.1.2 of [1]), S(1),S(z) and S(z2) are three roots of the cubic equation

    x33pxpc=0,

    where c is uniquely determined by

    4p=c2+27d2,  c1(mod3). (1.1)

    However, how to determine which of the three roots corresponds to S(1) is still an open problem.

    In this paper, for a fixed zGH, we find a relation between S(1),S(z) and S(z2).

    Theorem 1.1. Let p1(mod3) and zGH. Then

    S(1)=2pcos(θp), S(z)=2pcos(θpsgn(d)23π), S(z2)=2pcos(θp+sgn(d)23π),

    where θp=13arccos(c2p)+jp23π; jp is one of three values 1,0,1 and only dependent on p; c and d are uniquely determined by

    4p=c2+27d2,  c1(mod3),  9dc(2zp13+1)(modp). (1.2)

    Moreover, there is a unique multiplicative character χ of order 3 over Fp such that

    χ(z)=1+3i2, G(χ)=peisgn(d)θp.

    As application, we consider some congruence equations modp. For a1,a2,a3G, let M(a1,a2,a3) be the number of solutions of

    a1x31+a2x32+a3x330(modp),

    and let N(a1,a2,a3) be the number of solutions of

    a1x31+a2x32a3(modp).

    In [2], Chowla, Cowles and Cowles showed that M(1,1,1)=p2+c(p1). As pointed out in [3], the following is essentially included in the derivation of the cubic equation of periods by Gauss [4]: For a prime p1(mod3) and for zGH, then one has

    M(1,1,z)=p2+12(p1)(9dc),

    where c and d are uniquely determined by (1.1) (except for the sign of d).

    Chowla, Cowles and Cowles [3] determined the sign of d for the case 2GH as the following result shows.

    Proposition 1.2. [3] Let a prime p1(mod3). If 2GH, then for any zGH, one has

    M(1,1,z)=p2+12(p1)(9dc),

    where c and d are uniquely determined by (1.1) with

    dc(mod4)  for  z2(modH)

    and

    dc(mod4)  for  z4(modH).

    Recently, Hong and Zhu [7] solve the Gauss sign problem. In fact, they gave the following result.

    Proposition 1.3. [7] Let a prime p1(mod3) and zGH. Let g be a generator of the multiplicative group G. one has

    M(1,1,z)=p2+12(p1)(cδz(p)d),

    where c and d are uniquely determined by (1.1) with d>0 and

    δz(p)=(1)indg(d)3sgn(Im(r1+33r2i)).

    Here r1 and r2 are uniquely determined by

    4p=r21+27r22,  r11(mod3),  9r2(2gp13+1)r1(modp).

    Indeed, their result need to use the generator of group G (that is the primitive root of module p). However, for a large prime p, it is not easy to find the primitive root of module p. In this paper, we consider M(a1,a2,a3), N(a1,a2,a3) and determine the sign of d immediately by the coefficients a1,a2 and a3. We have the following three more general results.

    Theorem 1.4. Let a prime p1(mod3) and a1,a2,a3G.

    (1) For the case a1a2a3H, M(a1,a2,a3)=p2+c(p1);

    (2) For the case a1a2a3H, M(a1,a2,a3)=p2+12(p1)(9dc),

    where c and d are uniquely determined by

    4p=c2+27d2,  c1(mod3),  9dc(2(a1a2a3)p13+1)(modp). (1.3)

    Theorem 1.5. Let p1(mod3) and a1,a2,a3G.

    (1) For the case a1a2a3H,

    N(a1,a2,a3)={p2+c,ifa1a2(modH);p+1+c,otherwise.

    (2) For the case a1a2a3H,

    N(a1,a2,a3)={p2+12(9dc),ifa1a2(modH);p+1+12(9dc),otherwise,

    where c and d are uniquely determined by (1.3).

    Corollary 1.6. Let p1(mod3) and a1,a2,a3G. Then

    M(a1,a2,a3)c(a1a2a3)p13(modp).

    In [14], H. Zhang and W. P. Zhang proposed the following open problem:

    Can the number of solutions to the cubic congruence equation

    x31+x32+x33+x34z(modp) (1.4)

    be calculated when zG?

    Let L(z) be the number of solutions of the above Eq (1.4). In [12], W. P. Zhang and J. Y. Hu proved that

    L(z)={p36p12p(5c±27d),ifzGH;p36p+5cp,ifzH. (1.5)

    However, in [12], they also proposed an interesting open problem: How to determine the choice of sign in (1.5). In this paper, we solve the sign problem in (1.5), and get the following result.

    Theorem 1.7. Let p be a prime number and p1(mod3), let zGH. Then

    L(z)=p36p12p(5c27d),

    where c and d are uniquely determined by

    4p=c2+27d2,  c1(mod3),  9dc(2zp13+1)(modp).

    Lemma 2.1 (Theorem 3.1.3 of [1]). Let p1(mod3) and χ be a multiplicative character of order 3 over Fp. Then

    J(χ,χ)=c+33di2,

    where the Jacobi sum J(χ,χ)=p1a=1χ(a)χ(1a), c and d are uniquely determined by

    4p=c2+27d2,  c1(mod3),  9dc(2gp13+1)(modp)

    with g being the generator of the multiplicative group G of non-zero residues (modp) such that χ(g)=1+3i2.

    Lemma 2.2 (Lemma 4.1.1 of [1]). Let p1(mod3). Let g be a generator of the multiplicative group G of non-zero residues (modp) with χ(g)=1+3i2. Then

    G3(χ)=pJ(χ,χ).

    Lemma 2.3. Let p1(mod3) and zGH. Then there is a unique multiplicative character χ of order 3 over Fp such that

    χ(z)=1+3i2,  G3(χ)=pc+33di2,

    where c and d are uniquely determined by (1.2).

    Proof. Let g be a generator of the group G. Note that zGH. So we have indgz±1(mod3). If indgz1(mod3), we take g=g; If indgz1(mod3), we take g=(g)1. Hence g also is a generator of the group G and indgz1(mod3). Thus we have

    zp13(gindgz)p13gp13indgzgp13(modp).

    We take the multiplicative character χ()=e(indg()3). Obviously, we have

    χ(z)=e(indgz3)=e(13)=1+3i2=χ(g).

    Obviously, all of the multiplicative non-principal characters of order 3 over Fp are χ and ¯χ, ¯χ(z)=¯χ(z)=13i2. Thus χ is the unique multiplicative character of order 3 over Fp with χ(z)=1+3i2.

    Note that G3(χ)=pJ(χ,χ) by Lemma 2.2. Finally, using the Lemma 2.1, one immediately arrive the Lemma 2.3 as required.

    Lemma 2.4. Let χ be a multiplicative character of order 3. Then for any aG, we have

    S(a)=¯χ(a)G(χ)+χ(a)G(¯χ). (2.1)

    Proof. Let χ be any multiplicative character of order 3. Then we have

    1+χ(k)+¯χ(k)={3,ifkH;0,ifkGH.

    Thus for any aG, we have

    S(a)=p1k=0e(ak3/p)=1+p1k=1(1+χ(k)+¯χ(k))e(ak/p)=1+p1k=1e(ak/p)+p1k=1χ(k)e(ak/p)+p1k=1¯χ(k)e(ak/p)=¯χ(a)p1k=1χ(ak)e(ak/p)+χ(a)p1k=1¯χ(ak)e(ak/p)=¯χ(a)G(χ)+χ(a)G(¯χ).

    In this section, we prove Theorem 1.1. First, by Lemma 2.3, there is a unique multiplicative character χ of order 3 such that

    χ(z)=1+3i2,  G3(χ)=pc+33di2,

    where c and d are uniquely determined by (1.2). We can rewrite G3(χ) by argument, and get

    G3(χ)=p32e3iθsgn(d),

    where θ=13arccos(c2p). Thus we have

    G(χ)=pei(sgn(d)θ+j23π)=peisgn(d)(θ+sgn(d)j23π),

    where j is one of three values 1,0,1. Let jp=sgn(d)j. Thus we have

    G(χ)=peisgn(d)(θ+jp23π).

    Next, we will prove that jp does not depend on the sign of d. Note that G(¯χ)=χ(1)¯G(χ)=peisgn(d)(θ+jp23π). By Lemma 2.4, we have

    S(1)=¯χ(1)G(χ)+χ(1)G(¯χ)=G(χ)+G(¯χ)=2pcos[sgn(d)(θ+jp23π)]=2pcos(θ+jp23π).

    Obviously, by the definition of S(1), the value of S(1) doesn't depend on the sign of d. Thus we have that jp does not depend on the sign of d.

    Take θp=θ+jp23π. We have G(χ)=peisgn(d)θp and S(1)=2pcos(θp). By Lemma 2.4, we have

    S(z)=¯χ(z)G(χ)+χ(z)G(¯χ)=13i2peisgn(d)θp+1+3i2peisgn(d)θp=pei(sgn(d)θp2π3)+pei(sgn(d)θp2π3)=2pcos(sgn(d)θp2π3)=2pcos(θpsgn(d)2π3).

    Similarly, we have

    S(z2)=2pcos(θp+sgn(d)23π).

    This completes the proof of the Theorem 1.1.

    In this section, we prove Theorem 1.4, 1.5 and 1.7. First, we begin with the proof of Theorem 1.4.

    Proof of Theorem 1.4. By the orthogonality of additive character, we have

    M(a1,a2,a3)=1pp1m=0p1x1=0p1x2=0p1x3=0e(m(a1x31+a2x32+a3x33)p)=p2+1pp1m=1S(ma1)S(ma2)S(ma3).

    Then by Lemma 2.4, for any multiplicative character χ of order 3, we have

    M(a1,a2,a3)=p2+1pp1m=1[3j=1(¯χ(maj)G(χ)+χ(maj)G(¯χ))]=p2+1pp1m=1[¯χ(a1a2a3)G3(χ)+χ(a1a2a3)G3(¯χ)]+G(χ)(χ(¯a1¯a2a3)+χ(¯a1a2¯a3)+χ(a1¯a2¯a3))p1m=1¯χ(m)+G(¯χ)(χ(¯a1a2a3)+χ(a1¯a2a3)+χ(a1a2¯a3)))p1m=1χ(m)=p2+p1p[¯χ(a1a2a3)G3(χ)+χ(a1a2a3)G3(¯χ)].

    If a1a2a3H, thus we have χ(a1a2a3)=¯χ(a1a2a3)=1. Then by Lemma 2.3, we have

    M(a1,a2,a3)=p2+p1p(G3(χ)+G3(¯χ))=p2+(p1)[c+33di2+c33di2]=p2+c(p1).

    If a1a2a3GH, then by Lemma 2.3, we can take multiplicative character χ of order 3 satisfying

    χ(a1a2a3)=1+3i2,  G3(χ)=pc+33di2,

    where c and d are uniquely determined by (1.3). Thus we have

    M(a1,a2,a3)=p2+(p1)(13i2c+33di2+1+3i2c33di2)=p2+12(p1)(9dc).

    This completes the proof of the Theorem 1.4.

    Proof of Theorem 1.5. We have

    M(a1,a2,a3)=p1x1,x2,x3=0a1x31+a2x32+a3x330(modp)1=p1x3=1p1x1,x2=0a1x31+a2x32+a3x330(modp)1+p1x1,x2=0a1x31+a2x320(modp)1=p1x3=1p1x1,x2=0a1(x1¯x3)3+a2(x2¯x3)3a3(modp)1+1+p1x1=1p1x2=1(¯x1x2)3a1¯a2(modp)1=(p1)p1x1,x2=0a1x31+a2x32a3(modp)1+1+p1x1=1p1x=1x3a1¯a2(modp)1=(p1)N(a1,a2,a3)+1+p1x1=1p1x=1x3a1¯a2(modp)1.

    If a1a2(modH), the number of solutions of the congruence equation x3a1¯a2(modp) is exactly 3. Thus we have

    M(a1,a2,a3)=(p1)N(a1,a2,a3)+1+3(p1)=(p1)N(a1,a2,a3)+3p2.

    If a1a2(modH), the congruence equation x3a1¯a2(modp) has no solution. Thus we have

    M(a1,a2,a3)=(p1)N(a1,a2,a3)+1.

    Hence Theorem 1.5 immediately follows from Theorem 1.4.

    Proof of Theorem 1.7. First, by Lemma 2.3, there is a unique multiplicative character χ of order 3 such that

    χ(z)=1+3i2,  G3(χ)=pc+33di2,

    where c and d are uniquely determined by (1.2).

    Note that χ(1)=1. By the orthogonality of additive character and Lemma 2.3, we have

    L(z)=1pp1m=0p1x1=0p1x2=0p1x3=0p1x4=0e(m(x31+x32+x33+x34z)p)=p3+1pp1m=1S4(m)e(mzp)=p3+1pp1m=1[¯χ(m)G(χ)+χ(m)G(¯χ)]4e(mzp)=p36p+1pp1m=1[¯χ(m)G4(χ)+4pχ(m)G2(χ)+4p¯χ(m)G2(¯χ)+χ(m)G4(¯χ)]e(mzp)=p36p+1pG4(χ)p1m=1¯χ(m)e(mzp)+1pG4(¯χ)p1m=1χ(m)e(mzp)  +4G2(χ)p1m=1χ(m)e(mzp)+4G2(¯χ)p1m=1¯χ(m)e(mzp)=p36p+1pG4(χ)χ(z)G(¯χ)+1pG4(¯χ)¯χ(z)G(χ)+4¯χ(z)G3(χ)+4χ(z)G3(¯χ)=p36p+χ(z)G3(χ)+¯χ(z)G3(¯χ)+4¯χ(z)G3(χ)+4χ(z)G3(¯χ)=p36p+p1+3i2c+33di2+p13i2c33di2  +4p13i2c+33di2+4p1+3i2c33di2=p36p12p(5c27d).

    This completes the proof of the Theorem 1.7.

    Example 4.1. We take F31:={¯0,¯1,,¯30}. Consider the cubic equations x31+2x32+3x330(mod31) and x31+2x323(mod31).

    If the integers c and d satisfying that 431=c2+27d2,c1(mod3),9dc(2×63113+1)(mod31), then c=4,d=2. One can check that 231131(mod31) and 6311325(mod31), so 6 is not a cubic element in F31 and 2 is a cubic element in F31. Thus 6H and 12(modH).

    It then follows from Theorems 1.4 and 1.5 that the numbers M(1,2,3) and N(1,2,3) of the cubic equations x31+2x32+3x330(mod31) and x31+2x323(mod31) are given by

    M(1,2,3)=312+12(311)(9×24)=1171

    and

    N(1,2,3)=312+12(9×24)=36.

    We list the solutions of equation x31+2x323(mod31) as belove:

    (¯1,¯1);(¯1,¯5);(¯1,¯25);(¯5,¯1);(¯5,¯5);(¯5,¯25);(¯25,¯1);(¯25,¯5);(¯25,¯25);(¯6,¯4);(¯6,¯7);(¯6,¯20);(¯26,¯4);(¯26,¯7);(¯26,¯20);(¯30,¯4);(¯30,¯7);(¯30,¯20);(¯4,¯8);(¯4,¯9);(¯4,¯14);(¯7,¯8);(¯7,¯9);(¯7,¯14);(¯20,¯8);(¯20,¯9);(¯20,¯14);(¯16,¯17);(¯16,¯22);(¯16,¯23);(¯18,¯17);(¯18,¯22);(¯18,¯23);(¯28,¯17);(¯28,¯22);(¯28,¯23).

    The authors are partially supported by the National Natural Science Foundation of China (Grant No. 11871193, 12071132) and the Natural Science Foundation of Henan Province (No. 222300420493, 202300410031).



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