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
[1] | Mengke Lu, Shang Gao, Xibei Yang, Hualong Yu . Improving performance of decision threshold moving-based strategies by integrating density-based clustering technique. Electronic Research Archive, 2023, 31(5): 2501-2518. doi: 10.3934/era.2023127 |
[2] | Kai Huang, Chang Jiang, Pei Li, Ali Shan, Jian Wan, Wenhu Qin . A systematic framework for urban smart transportation towards traffic management and parking. Electronic Research Archive, 2022, 30(11): 4191-4208. doi: 10.3934/era.2022212 |
[3] | Ilyоs Abdullaev, Natalia Prodanova, Mohammed Altaf Ahmed, E. Laxmi Lydia, Bhanu Shrestha, Gyanendra Prasad Joshi, Woong Cho . Leveraging metaheuristics with artificial intelligence for customer churn prediction in telecom industries. Electronic Research Archive, 2023, 31(8): 4443-4458. doi: 10.3934/era.2023227 |
[4] | Dong-hyeon Kim, Se-woon Choe, Sung-Uk Zhang . Recognition of adherent polychaetes on oysters and scallops using Microsoft Azure Custom Vision. Electronic Research Archive, 2023, 31(3): 1691-1709. doi: 10.3934/era.2023088 |
[5] | Qing Tian, Heng Zhang, Shiyu Xia, Heng Xu, Chuang Ma . Cross-view learning with scatters and manifold exploitation in geodesic space. Electronic Research Archive, 2023, 31(9): 5425-5441. doi: 10.3934/era.2023275 |
[6] | Nihar Patel, Nakul Vasani, Nilesh Kumar Jadav, Rajesh Gupta, Sudeep Tanwar, Zdzislaw Polkowski, Fayez Alqahtani, Amr Gafar . F-LSTM: Federated learning-based LSTM framework for cryptocurrency price prediction. Electronic Research Archive, 2023, 31(10): 6525-6551. doi: 10.3934/era.2023330 |
[7] | Xiaoyan Wu, Guowen Ye, Yongming Liu, Zhuanzhe Zhao, Zhibo Liu, Yu Chen . Application of Improved Jellyfish Search algorithm in Rotate Vector reducer fault diagnosis. Electronic Research Archive, 2023, 31(8): 4882-4906. doi: 10.3934/era.2023250 |
[8] | Shizhen Huang, Enhao Tang, Shun Li, Xiangzhan Ping, Ruiqi Chen . Hardware-friendly compression and hardware acceleration for transformer: A survey. Electronic Research Archive, 2022, 30(10): 3755-3785. doi: 10.3934/era.2022192 |
[9] | Youqun Long, Jianhui Zhang, Gaoli Wang, Jie Fu . Hierarchical federated learning with global differential privacy. Electronic Research Archive, 2023, 31(7): 3741-3758. doi: 10.3934/era.2023190 |
[10] | Duhui Chang, Yan Geng . Distributed data-driven iterative learning control for multi-agent systems with unknown input-output coupled parameters. Electronic Research Archive, 2025, 33(2): 867-889. doi: 10.3934/era.2025039 |
For a prime p≡1(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 a∈G, we defined the sums
S(a)=p−1∑k=0e(ak3/p) |
and
G(χ)=p−1∑k=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 z∈G∖H. 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
x3−3px−pc=0, |
where c is uniquely determined by
4p=c2+27d2, c≡1(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 z∈G∖H, we find a relation between S(1),S(z) and S(z2).
Theorem 1.1. Let p≡1(mod3) and z∈G∖H. Then
S(1)=2√pcos(θp), S(z)=2√pcos(θp−sgn(d)23π), S(z2)=2√pcos(θp+sgn(d)23π), |
where θp=13arccos(−c2√p)+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, c≡1(mod3), 9d≡c(2zp−13+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,a3∈G, let M(a1,a2,a3) be the number of solutions of
a1x31+a2x32+a3x33≡0(modp), |
and let N(a1,a2,a3) be the number of solutions of
a1x31+a2x32≡a3(modp). |
In [2], Chowla, Cowles and Cowles showed that M(1,1,1)=p2+c(p−1). 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 p≡1(mod3) and for z∈G∖H, then one has
M(1,1,z)=p2+12(p−1)(9d−c), |
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 2∈G∖H as the following result shows.
Proposition 1.2. [3] Let a prime p≡1(mod3). If 2∈G∖H, then for any z∈G∖H, one has
M(1,1,z)=p2+12(p−1)(9d−c), |
where c and d are uniquely determined by (1.1) with
d≡c(mod4) for z≡2(modH) |
and
d≡−c(mod4) for z≡4(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 p≡1(mod3) and z∈G∖H. Let g be a generator of the multiplicative group G. one has
M(1,1,z)=p2+12(p−1)(−c−δz(p)d), |
where c and d are uniquely determined by (1.1) with d>0 and
δz(p)=(−1)⟨indg(d)⟩3⋅sgn(Im(r1+3√3r2i)). |
Here r1 and r2 are uniquely determined by
4p=r21+27r22, r1≡1(mod3), 9r2≡(2gp−13+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 p≡1(mod3) and a1,a2,a3∈G.
(1) For the case a1a2a3∈H, M(a1,a2,a3)=p2+c(p−1);
(2) For the case a1a2a3∉H, M(a1,a2,a3)=p2+12(p−1)(9d−c),
where c and d are uniquely determined by
4p=c2+27d2, c≡1(mod3), 9d≡c(2(a1a2a3)p−13+1)(modp). | (1.3) |
Theorem 1.5. Let p≡1(mod3) and a1,a2,a3∈G.
(1) For the case a1a2a3∈H,
N(a1,a2,a3)={p−2+c,ifa1≡a2(modH);p+1+c,otherwise. |
(2) For the case a1a2a3∉H,
N(a1,a2,a3)={p−2+12(9d−c),ifa1≡a2(modH);p+1+12(9d−c),otherwise, |
where c and d are uniquely determined by (1.3).
Corollary 1.6. Let p≡1(mod3) and a1,a2,a3∈G. Then
M(a1,a2,a3)≡−c(a1a2a3)p−13(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+x34≡z(modp) | (1.4) |
be calculated when z∈G?
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)={p3−6p−12p(5c±27d),ifz∈G∖H;p3−6p+5cp,ifz∈H. | (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 p≡1(mod3), let z∈G∖H. Then
L(z)=p3−6p−12p(5c−27d), |
where c and d are uniquely determined by
4p=c2+27d2, c≡1(mod3), 9d≡c(2zp−13+1)(modp). |
Lemma 2.1 (Theorem 3.1.3 of [1]). Let p≡1(mod3) and χ be a multiplicative character of order 3 over Fp. Then
J(χ,χ)=c+3√3di2, |
where the Jacobi sum J(χ,χ)=∑p−1a=1χ(a)χ(1−a), c and d are uniquely determined by
4p=c2+27d2, c≡1(mod3), 9d≡c(2gp−13+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 p≡1(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 p≡1(mod3) and z∈G∖H. Then there is a unique multiplicative character χ of order 3 over Fp such that
χ(z)=−1+√3i2, G3(χ)=p⋅c+3√3di2, |
where c and d are uniquely determined by (1.2).
Proof. Let g′ be a generator of the group G. Note that z∈G∖H. So we have indg′z≡±1(mod3). If indg′z≡1(mod3), we take g=g′; If indg′z≡−1(mod3), we take g=(g′)−1. Hence g also is a generator of the group G and indgz≡1(mod3). Thus we have
zp−13≡(gindgz)p−13≡gp−13indgz≡gp−13(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)=−1−√3i2. 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 a∈G, 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,ifk∈H;0,ifk∈G∖H. |
Thus for any a∈G, we have
S(a)=p−1∑k=0e(ak3/p)=1+p−1∑k=1(1+χ(k)+¯χ(k))e(ak/p)=1+p−1∑k=1e(ak/p)+p−1∑k=1χ(k)e(ak/p)+p−1∑k=1¯χ(k)e(ak/p)=¯χ(a)p−1∑k=1χ(ak)e(ak/p)+χ(a)p−1∑k=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)=p⋅c+3√3di2, |
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(−c2√p). 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)=√pe−isgn(d)(θ+jp23π). By Lemma 2.4, we have
S(1)=¯χ(1)G(χ)+χ(1)G(¯χ)=G(χ)+G(¯χ)=2√pcos[sgn(d)(θ+jp23π)]=2√pcos(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)=2√pcos(θp). By Lemma 2.4, we have
S(z)=¯χ(z)G(χ)+χ(z)G(¯χ)=−1−√3i2⋅√peisgn(d)θp+−1+√3i2⋅√pe−isgn(d)θp=√pei(sgn(d)θp−2π3)+√pe−i(sgn(d)θp−2π3)=2√pcos(sgn(d)θp−2π3)=2√pcos(θp−sgn(d)2π3). |
Similarly, we have
S(z2)=2√pcos(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)=1pp−1∑m=0p−1∑x1=0p−1∑x2=0p−1∑x3=0e(m(a1x31+a2x32+a3x33)p)=p2+1pp−1∑m=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+1pp−1∑m=1[3∏j=1(¯χ(maj)G(χ)+χ(maj)G(¯χ))]=p2+1pp−1∑m=1[¯χ(a1a2a3)G3(χ)+χ(a1a2a3)G3(¯χ)]+G(χ)(χ(¯a1¯a2a3)+χ(¯a1a2¯a3)+χ(a1¯a2¯a3))p−1∑m=1¯χ(m)+G(¯χ)(χ(¯a1a2a3)+χ(a1¯a2a3)+χ(a1a2¯a3)))p−1∑m=1χ(m)=p2+p−1p[¯χ(a1a2a3)G3(χ)+χ(a1a2a3)G3(¯χ)]. |
If a1a2a3∈H, thus we have χ(a1a2a3)=¯χ(a1a2a3)=1. Then by Lemma 2.3, we have
M(a1,a2,a3)=p2+p−1p(G3(χ)+G3(¯χ))=p2+(p−1)[c+3√3di2+c−3√3di2]=p2+c(p−1). |
If a1a2a3∈G∖H, then by Lemma 2.3, we can take multiplicative character χ of order 3 satisfying
χ(a1a2a3)=−1+√3i2, G3(χ)=p⋅c+3√3di2, |
where c and d are uniquely determined by (1.3). Thus we have
M(a1,a2,a3)=p2+(p−1)(−1−√3i2⋅c+3√3di2+−1+√3i2⋅c−3√3di2)=p2+12(p−1)(9d−c). |
This completes the proof of the Theorem 1.4.
Proof of Theorem 1.5. We have
M(a1,a2,a3)=p−1∑x1,x2,x3=0a1x31+a2x32+a3x33≡0(modp)1=p−1∑x3=1p−1∑x1,x2=0a1x31+a2x32+a3x33≡0(modp)1+p−1∑x1,x2=0a1x31+a2x32≡0(modp)1=p−1∑x3=1p−1∑x1,x2=0a1(−x1¯x3)3+a2(x2¯x3)3≡a3(modp)1+1+p−1∑x1=1p−1∑x2=1(−¯x1x2)3≡a1¯a2(modp)1=(p−1)p−1∑x1,x2=0a1x31+a2x32≡a3(modp)1+1+p−1∑x1=1p−1∑x=1x3≡a1¯a2(modp)1=(p−1)N(a1,a2,a3)+1+p−1∑x1=1p−1∑x=1x3≡a1¯a2(modp)1. |
If a1≡a2(modH), the number of solutions of the congruence equation x3≡a1¯a2(modp) is exactly 3. Thus we have
M(a1,a2,a3)=(p−1)N(a1,a2,a3)+1+3(p−1)=(p−1)N(a1,a2,a3)+3p−2. |
If a1≢a2(modH), the congruence equation x3≡a1¯a2(modp) has no solution. Thus we have
M(a1,a2,a3)=(p−1)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)=p⋅c+3√3di2, |
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)=1pp−1∑m=0p−1∑x1=0p−1∑x2=0p−1∑x3=0p−1∑x4=0e(m(x31+x32+x33+x34−z)p)=p3+1pp−1∑m=1S4(m)e(−mzp)=p3+1pp−1∑m=1[¯χ(m)G(χ)+χ(m)G(¯χ)]4e(−mzp)=p3−6p+1pp−1∑m=1[¯χ(m)G4(χ)+4pχ(m)G2(χ)+4p¯χ(m)G2(¯χ)+χ(m)G4(¯χ)]e(−mzp)=p3−6p+1pG4(χ)p−1∑m=1¯χ(m)e(−mzp)+1pG4(¯χ)p−1∑m=1χ(m)e(−mzp) +4G2(χ)p−1∑m=1χ(m)e(−mzp)+4G2(¯χ)p−1∑m=1¯χ(m)e(−mzp)=p3−6p+1pG4(χ)χ(−z)G(¯χ)+1pG4(¯χ)¯χ(−z)G(χ)+4¯χ(−z)G3(χ)+4χ(−z)G3(¯χ)=p3−6p+χ(z)G3(χ)+¯χ(z)G3(¯χ)+4¯χ(z)G3(χ)+4χ(z)G3(¯χ)=p3−6p+p⋅−1+√3i2⋅c+3√3di2+p⋅−1−√3i2⋅c−3√3di2 +4p⋅−1−√3i2⋅c+3√3di2+4p⋅−1+√3i2⋅c−3√3di2=p3−6p−12p(5c−27d). |
This completes the proof of the Theorem 1.7.
Example 4.1. We take F31:={¯0,¯1,⋯,¯30}. Consider the cubic equations x31+2x32+3x33≡0(mod31) and x31+2x32≡3(mod31).
If the integers c and d satisfying that 4⋅31=c2+27d2,c≡1(mod3),9d≡c(2×631−13+1)(mod31), then c=4,d=2. One can check that 231−13≡1(mod31) and 631−13≡25(mod31), so 6 is not a cubic element in F31 and 2 is a cubic element in F31. Thus 6∉H and 1≡2(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+3x33≡0(mod31) and x31+2x32≡3(mod31) are given by
M(1,2,3)=312+12(31−1)(9×2−4)=1171 |
and
N(1,2,3)=31−2+12(9×2−4)=36. |
We list the solutions of equation x31+2x32≡3(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).
[1] | M. Aschenbrenner, L. Van Den Dries, J. Van Der Hoeven, Asymptotic Differential Algebra and Model Theory of Transseries, Princeton University Press, 2017. |
[2] | A. G. Czerniakiewicz, Automorphisms of a free associative algebra of rank 2. I & II, T. Am. Math. Soc., 160 (1971), 393-401; 171 (1972), 309-315. |
[3] |
P. M. Cohn, Subalgebras of free associative algebras, P. Lond. Math. Soc., 56 (1964), 618-632. doi: 10.1112/plms/s3-14.4.618
![]() |
[4] | B. A. Duisengaliyeva, A. S. Naurazbekova, U. U. Umirbaev, Tame and wild automorphisms of differential polynomial algebras of rank 2, Fund. Appl. Math., 22 (2019), 101-114. |
[5] | G. Gallo, B. Mishra, F. Ollivier, Some constructions in rings of differential polynomials, In: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Springer, Berlin, 1991, 171-182. |
[6] | H. W. E. Jung, Uber ganze birationale Transformationen der Ebene, J. Reine Angew. Math., 184 (1942), 161-174. |
[7] | I. Kaplansky, An Introduction to Differential Algebra, Hermann, Paris, 1957. |
[8] | E. R. Kolchin, Differential Algebra and Algebraic Groups, Academic Press, New York, 1973. |
[9] | W. van der Kulk, On polynomial rings in two variables, Nieuw Archief voor Wiskunde, 3 (1953), 33-41. |
[10] |
L. Makar-Limanov, The automorphisms of the free algebra with two generators, Funct. Anal. Appl., 4 (1970), 262-263. doi: 10.1007/BF01075252
![]() |
[11] | M. Nagata, On the Automorphism Group of k[x,y], Kinokuniya, Tokio, 1972. |
[12] | J. F. Ritt, Differential Algebra, American Mathematical Society, New York, 1950. |
[13] |
I. P. Shestakov, U. U. Umirbaev, Tame and wild automorphisms of rings of polynomials in three variables, J. Am. Math. Soc., 17 (2004), 197-227. doi: 10.1090/S0894-0347-03-00440-5
![]() |
[14] | W. Sit, The Ritt-Kolchin Theory for Differential Polynomials, World Scientific, Singapore, 2001. |
[15] | U. U. Umirbaev, The Anick automorphism of free associative algebras, J. Reine Angew. Math., 605 (2007), 165-178. |
[16] | A. Van den Essen, Polynomial Automorphisms: and the Jacobian Conjecture, Birkhauser, 2012. |
1. | Rufan Lin, Yongkang Chen, Lekai Qiu, Yihan Yu, Fan Xia, The Influence of Interactivity, Aesthetic, Creativity and Vividness on Consumer Purchase of Virtual Clothing: The Mediating Effect of Satisfaction and Flow, 2024, 1044-7318, 1, 10.1080/10447318.2024.2359226 | |
2. | Xubing Xu, Qiong Luo, Tian Zhong, Research on innovative visualization design of miao costume images in Qiandongnan under aesthetic perspective, 2024, 9, 2444-8656, 10.2478/amns-2024-2727 | |
3. | Zhengtang Tan, Shuang Lin, Zebin Wang, Cluster Size Intelligence Prediction System for Young Women’s Clothing Using 3D Body Scan Data, 2024, 12, 2227-7390, 497, 10.3390/math12030497 | |
4. | Gu Xiaoxue, 2024, Research on Application System of Computer Aided Design in Innovative Design of Intangible Cultural Heritage Clothing, 979-8-3503-6024-0, 1993, 10.1109/ICIPCA61593.2024.10709043 | |
5. | Yi Xiang, 2024, Intelligent Clothing Design and Embodied Cognitive System Based on Convolutional Neural Network and Generative Adversarial Network, 979-8-3315-2762-4, 1173, 10.1109/ICEDCS64328.2024.00214 | |
6. | Meizhen Deng, Ling Chen, CDGFD: Cross-Domain Generalization in Ethnic Fashion Design Using LLMs and GANs: A Symbolic and Geometric Approach, 2025, 13, 2169-3536, 7192, 10.1109/ACCESS.2024.3524444 | |
7. | Miao Yu, Geometric modeling and computer-aided creation methods in traditional cultural costume design, 2025, 10, 2444-8656, 10.2478/amns-2025-0717 | |
8. | Jingting Meng, Xingjia Fang, Jian Xu, Ziqi Zhang, Research on the Innovative Application of Song Dynasty Boundary Painting in Interior Soft Decoration Design Based on AIGC, 2025, 15, 2075-5309, 1067, 10.3390/buildings15071067 |