Tests of Non-Equivalence among Absolutely Nonsingular Tensors through Geometric Invariants

T ests of Non-Equiv alence among Absolutely Nonsingular T ensors through Geometric In v arian ts Sak ata, T. 1 , Maehra, K. 2 , Sasaki,T. 3 , Sumi, T. 1 , Miyazaki, M. 4 and W atanab e Y. 5 Departmen t of Design Human Science, Kyushu Univ ersity 1 Sc ho ol of Design, Kyush u Univ ersit y 2 . Departmen t of Mathematics, Kob e Universit y 3 . Departmen t of Mathematics, Ky oto Univ ersit y of Education 4 Researc h Institute of Information T echnology , Kyush u Univ ersity 5 1 In tro duction T ensor data analysis has successfully developed in v arious application fields, whic h is useful to seize m ulti-factor dependence. An n × n × p tensor is a m ulti-arra y datum T = ( T ij k ) , where 1 ≤ i, j ≤ n and 1 ≤ k ≤ p. A type n × n × p tensor T is denoted b y T = ( A 1 ; A 2 ; · · · ; A p ) where A i denote n × n matrices. An n × n × p tensor T is said to b e of rank 1 if there is a v ectors a = ( a 1 , a 2 , ..., a n ) , b = ( b 1 , b 2 , ..., b n ), and c = ( c 1 , c 2 , ..., c p ) suc h that T ij k = a i b j c k for all i, j, k , and the rank of a tensor T is defined as the minim um of the in teger r such that T can b e expressed as the sum of r rank-one tensors. The maximal rank of all tensors of t yp e n × n × p are also defined in ob vious fashion and denoted b y maxrank(n , n , p).The rank of a tensor describ es the complexity of a tensorial datum and the maximal rank describ es the mo del complexity of a class of tensors of a giv en type, and so they are v ery imp ortan t concepts in both applied and theoretical fields. Therefore, the rank and maximal rank determination problems ha ve attracted the interest of man y researc hers, for example, Krusk al [12], ten-Berge [24], and Common et al, [6], etc. and no w ha v e b eing in vestigated in tensiv ely( for a comprehensive surv ey , see Kolda et al.[11]). A tkinson et al.([1] and [2] ) claimed that maxrank( n, n, 3) ≤ 2 n − 1. Here w e in tro duce an imp ortan t class of tensors. Definition 1.1 A r e al tensor T = ( A 1 ; A 2 ; , . . . ; A p ) is said to b e absolutely nonsin- gular if x 1 A 1 + x 2 A 2 + · · · + x p A p is nonsingular for al l ( x 1 , x 2 , ..., x p ) 6 = (0 , 0 , ..., 0) . Remark 1.2 We c al le d absolutely nonsingular tensors as exc eptional tensors in Sakata et al. [21],[22] . Sumi et al.[23] prov ed the claim of Atkinson et al. o ver the complex n umber filed C without an y assumption and pro v ed it o ver the real n umber filed R except the class of absolutely nonsingular tensors. Thus, for the pro of of the claim of A tkinson et al. o ver the real n umber filed R , it is the first thing to determine all absolutely nonsingular tensors. Absolutely nonsingular tensors are c haracterized by the de- terminan t p olynomial defined b elow. Searc hing of absolutely nonsingular tensors w as pursued in Sak ata et al. [21],[22] in this direction. As w ell as searc hing abso- lutely nonsingular tensors, the equiv alence among them under the rank-preserving transformation whic h is defined below is also imp ortan t. Note that such equiv alence relation has also a relation to the SLOCC equiv alence of entangled states in the quan tum comm unication (for example, see Chen et al. [5]). 1 Definition 1.3 F or a n × n × p tensor T = ( A 1 ; A 2 ; , . . . ; A p ) , the homo gene ous p olynomial in x 1 , ..., x p of de gr e e n f T ( x 1 , x 2 , ..., x p ) = det( x 1 A 1 + x 2 A 2 + · · · + x p A p ) (1.4) is c al le d the determinant p olynomial of a tensor T . Then w e hav e the following imp ortan t c haracterization. Theorem 1.5 If T = ( A 1 ; A 2 ; , . . . ; A p ) is absolutely nonsingular, its determinant p olynomial f T ( x 1 , x 2 , ..., x p ) is a p ositive definite homo gene ous p olynomial or ne gative definite homo gene ous p olynomial. Pro of Let T = ( A 1 ; A 2 ; , . . . ; A p ) b e absolutely nonsingular, and assume that there are t w o p oin ts x 0 and x 1 suc h that f T ( x 0 ) > 0 and f T ( x 1 ) < 0 . The line ` com bining the t wo p oin ts x 0 and x 1 m ust pass through the origin 0 , since in the segmen t [ x 0 , x 1 ] there m ust b e x 0 suc h that f T ( x 0 ) = 0 and it m ust b e 0 b ecause T is absolutely nonsingular. Let tak e another p oin t x 2 whic h is not on the line ` . Then, the line passing x 0 and x 2 do es not pass the origin and so f ( x 0 ) f ( x 2 ) < 0 is imp ossible just b y the same reason giv en in the previous sen tence. So, f ( x 0 ) f ( x 2 ) > 0 . Next, consider the line passing x 1 and x 2 , whic h also do es not pass the origin and f ( x 1 ) f ( x 2 ) < 0. This is also a con tradiction. After all, there don’t exist points x 0 and x 1 suc h that f ( x 0 ) > 0 and f ( x 1 ) < 0 . This pro v es Theorem 1.5. It is w ell kno wn that tensor rank is in v ariant b y t ypical matrix transformations, sa y , p − , q − , and r − transformations defined below. So, equiv alence relation of tw o tensors means that they ha ve a same rank. Th us, to study equiv alence among tensors is of some imp ortance for rank determination. Definition 1.6 F or a n × n × p tensor T = ( A 1 ; A 2 ; · · · ; A p ) , the fol lowi ng tr ans- formations (1) T = ( A 1 ; A 2 ; · · · ; A p ) → T 0 = ( P A 1 ; P A 2 ; · · · ; P A p ) by an n × n matrix P ∈ GL ( n ) , (2) T = ( A 1 ; A 2 ; · · · ; A p ) → T 0 = ( A 1 Q ; P A 2 Q ; · · · ; A p Q ) by an n × n matrix P ∈ GL ( n ) , (3) T = ( A 1 ; A 2 ; · · · ; A p ) → T 0 = ( R 11 A 1 + R 12 A 2 + R 13 A 3 ; R 21 A 1 + R 22 A 2 + R 23 A 3 ; R 31 A 1 + R 32 A 2 + R 33 A 3 ) by a p × p matrix P ∈ GL ( n ) ar e c al le d as p − , q − , and r − tr ansformations and denote d by T → p T 0 , T → q T 0 andT → r T 0 r esp e ctively. F urther, if T 1 → p T 2 , the T 1 and T 2 ar e said to b e in the p − e quivalenc e. q − and r − e quivalenc e ar e define d analo gously. Definition 1.7 L et T 1 = ( A 1 ; A 2 ; · · · ; A p ) and T 2 = ( B 1 ; B 2 ; · · · ; B p ) b e two n × n × p tensors. If ther e is a se quenc e of { T i } starting fr om T 1 and ending at T 2 , in which T i and T i +1 ar e in the r elation of p − , or q − , or r − e quivalenc e, then T 0 and T 1 ar e said to b e e quivalent. 2 No w w e can reduce the equiv alence relation into a more simple one by the following lemma. Lemma 1.8 p − , q − and r − tr ansformations ar e mutual ly c ommutative. Pro of F or simplicity , we prov e for p = 3, ho w ever, the pro of is similar for a general p . First w e pro v e the commutativit y of p − transformation and r − transformation. Let T 1 → p T 2 → r T 3 and T 1 → r T 0 2 → p T 0 3 W e will sho w that T 3 = T 0 3 . Let T 1 = ( A 1 ; A 2 ; A 3 ) and P = ( p ij ) and R = ( r ij ) . Then, T 2 = ( P A 1 ; P A 2 ; P A 3 ) and T 3 = ( r 11 P A 1 + r 12 P A 2 + r 13 P A 3 ; r 21 P A 1 + r 22 P A 2 + r 23 P A 3 ; r 31 P A 1 + r 32 P A 2 + r 33 P A 3 ) On the other hand T 0 2 = ( r 11 A 1 + r 12 A 2 + r 13 A 3 ; r 21 A 1 + r 22 A 2 + r 23 A 3 ; r 31 A 1 + r 32 A 2 + r 33 A 3 ) and T 0 3 = ( r 11 P A 1 + r 12 P A 2 + r 13 P A 3 ; r 21 P A 1 + r 22 P A 2 + r 23 P A 3 ; r 31 P A 1 + r 32 P A 2 + r 33 P A 3 ) Th us, T 3 = T 0 3 , and this means the commutativit y of p - and r -transformations. The commutativit y of q - and r -transformations are pro ved similarly . p - and q - transformations are obviously comm utative. This pro v es Lemma 1.8. Note that in this pap er we consider three cases of (1) P , Q ∈ GL ( n ) and R ∈ GL ( p ) and (2) P , Q ∈ GL ( n ) and R ∈ S L ( p ) . and (3) P , Q ∈ S L ( n ) and R ∈ S L ( p ) . The first is called GL ( p )-equiv alence or simply equiv alence, and the second is called S L ( p )-equiv alence in short. The third case is called, in a full term, S L ( n ) × S L ( n ) × S L ( p )-equiv alence. Lemma 1.8 implies the follo wing theorem. Theorem 1.9 T 1 and T 2 ar e GL ( p ) -e quivalent if and only if ther e is a set of p - tr ansformation, q -tr ansformation and r -tr ansformation such that T 1 → p T 0 → q T ” → r T 2 Th us, the equiv alence problem of tensors T 1 = ( A (1) 1 : , , , : A (1) p ) and T 2 = ( A (2) 1 : , , , : A (2) p ) is reduced to the problem whether the following system of algebraic equations for P , Q and R can hav e a solution or not. A (2) i = P ( { p X j =1 r ij A (1) j } Q, i = 1 , 2 , ..., p (1.10) These algebraic equations ha ve to o man y v ariables to solv e ev en when the size of matrices A i is mo derate. So, in this pap er, w e prop ose to see the problem through the determinan t polynomial. Then, though we necessarily hav e to discard the sufficiency part of the problem, how ev er, the problem b ecomes concise and tractable one by the follo wing prop osition. 3 Prop osition 1.11 If T 1 and T 2 ar e GL ( p ) -e quivalent, it holds that ther e is a c on- stant c ∈ R and a p × p nonsingular matrix R ∈ GL ( p ) such that f T 2 ( x ) = cf T 1 ( x R ) (1.12) So, w e can sa y that Prop osition 1.13 F or two tensors, if the e quation (1.12) do es not hold for any c onstant c ∈ R and any matrix R ∈ GL ( p ) , they ar e not GL ( p ) -e quivalent. Though the reduced equation (1.12) happ ens to b e solved algebraically in some cases. How ev er, it is still hard to solv e, in general, a system of algebraic equation with to o man y v ariables. In fact, we need to decide whether a system of ( n +1)( n +2) 2 homogeneous equations with ( p 2 + 1) v ariables of degree n hav e a solution or not. So, in this pap er, w e av oid to solve the problem algebraically and propose to attack the problem from a geometric view p oin t, that is, w e propose to test non equiv alence b y chec king whether the t wo surfaces of the determinan t p olynomials of T 1 and T 2 ha ve a same geometric in v ariants, or not. Here, m ulti-linear algebra and differential geometry in tersect through the wido w of determinan t p olynomials. The first aim of this pap er is to show theoretically that differential geometric in v ari- an ts are useful as testers of non-equiv alence among absolutely nonsingular tensors. The second aim is to show that w e can calculate the v alues of the in v arian ts with enough accuracy . Third, we compare the v alues of in v arian ts calculated b y the lat- tice metho d and b y the t-design metho d. And it is shown that the lattice p oin t metho d giv es more stable v alues than the t-design metho d. As S L ( p )-inv arian t, w e consider first the volume enclosed b y the constan t surface and then we consider the affine surface area, and thirdly w e consider the L p affine surface area of con v ex b ody . Affine surface area w as studied b y Blasc hk e [4] and extended to L p affine surface area b y Lut w ak [20], (also see Leich t wess [13]). As for a v aluation theory of L p affine surface area, see the recen t papers b y Ludwig [15] and Ludwig and Reitzer [17]. Finally , as a general reference of affine differential geometry , see K. Nomizu and T. Sasaki [19]. This pap er is organized as follo ws. In Section 2, w e show how to parametrize the constan t surface of a determinan t p olynomial and in Section 3, w e review briefly some definitions from differential geometry . In Section 4, w e argue rough S L ( p )- in v arian ts. In Section 5, we deal with S L ( p )-inv ariant. In the first subsection, w e in tro duce the v aluation theory for the set of con v ex b odies and in the second subsection, w e argue a v olume of the region enclosed b y a constant surface as an S L ( p )-inv ariant. In the third subsection, we argue the affine surface as a S L ( p )- in v arian t. In Section 6, w e consider the generalized affine surface, that is, L p affine surface area, esp ecially cen tro-affine surface as a GL ( p )-in v arian t. In Section 7, we review the theory of spherical t-design briefly and give a theorem imp ortan t for ap- pro ximate calculation of our prop osed in v arian ts. In Section 8, w e give n umerical v alues of the inv ariants calculated by the lattice metho d and t-design metho d. It is sho wn n umerically that the prop osed in v arian ts is usefull to discriminate non equiv- alence. In Section 9, the conclusion is given. Finally note that in the follo wing we consider mainly the case of n = 4 and p = 3 , though some statements are giv en for general n and p. One reason is that absolutely nonsingular tensors are not so easy 4 to obtain for general cases and the second reason is b ecause it is easy to see that our metho d is also a v ailable for general cases. The study of m uch higher v alues of n and p will b e giv en in the future work. 2 P arametrization of constan t surface The determinan t p olynomial of a 4 × 4 × 3 tensor T = ( A 1 ; A 2 ; A 3 ), i f T ( x, y , z ) = det ( xA 1 + y A 2 + z A 3 ) , is a homogeneous polynomial of three v ariables with degree 4. W e are concerned with the integral in v arian ts of the constant surface ∂ Ω T = { ( x, y , z ) | f T ( x, y , z ) = 1 } for the special linear group S L (3) and the general linear group GL (3). T o get such inv arian ts, we need to parametrize this surface by the usual spherical co ordinate, x = r sin s cos t = r Φ x ( s, t ) (2.1) y = r sin s sin t = r Φ y ( s, t ) (2.2) z = r cos s = r Φ z ( s, t ) , (2.3) where 0 < s < π , 0 < t < 2 π . Let x denote the point ( x, y , z ) on the surface. Putting these in to the equation f T ( x, y , z ) = 1 , we hav e r 4 = 1 p ( s, t ) , (2.4) where p ( s, t ) = f T (Φ x ( s, t ) , Φ y ( s, t ) , Φ z ( s, t )) . (2.5) And so, x = 1 p ( s, t ) 1 / 4 (Φ x ( s, t ) , Φ y ( s, t ) , Φ z ( s, t )) . (2.6) This equation (2.6) giv es a parametric represen tation of the constant surface ∂ Ω T . Then, the following is a starting p oin t of this researc h of the constan t surface. Theorem 2.7 The c onstant surfac e of the determinant p olynomial of an absolutely nonsingular tensor is a c omp act set in R 3 without self-interse ction. Pro of Without loss of generalit y , we assume that f T ( x ) is p ositiv e definite. If x ∈ ∂ Ω T , for an y 0 < r < 1 and r > 1 , r x is not in ∂ Ω T . That is, the constan t surface is of a star-shap ed. This prov es that the surface has not an y self intersection. Since p ( s, t ) is con tinuous on the unit sphere it takes a p ositiv e minimum and a p ositive maxim um. So, x in the equation 2.6 is b ounded, whic h implies the compactness of the constan t surface. This completes the proof of Theorem 2.7. The follo wing 8 figures are examples of the constant surfaces of 4 × 4 × 3 absolutely nonsingular tensors. Note that the n umbering of tensors is based on the list of nonsingular tensors with elements consisting only of − 1 , 0 , 1 found by us. Each figure corresponds to 5 Figure 1: Constan t surfaces of the determinant polynomials of tensorsF N o 1 , 3 , 10 , 20 , 99 , 119 , 207 , 237 . the follo wing tensor and its determinant function respectively . T 1 =     1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1     :     − 1 0 1 1 1 − 1 1 − 1 − 1 − 1 − 1 − 1 0 1 1 0     :     0 1 − 1 1 0 − 1 1 1 0 − 1 0 − 1 − 1 − 1 1 1     , f T 1 ( x, y , z ) = x 4 + 6 y 4 + 2 z 4 + − 3 x 3 y − 8 xy 3 − 3 xz 3 + 5 z 3 y + 7 x 2 y 2 + 3 x 2 z 2 + 8 z 2 y 2 − 2 xy 2 z − 8 xy z 2 . T 3 =     1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1     :     0 1 0 0 0 − 1 0 1 1 0 − 0 − 1 1 − 1 1     :     1 1 1 1 − 1 0 0 − 1 0 − 1 0 0 − 1 1 − 1 1     , f T 3 ( x, y , z ) = x 4 + 3 y 4 + 6 z 4 − 3 x 3 z + 2 xy 3 + 4 y 3 z − 7 xz 3 − 6 z 3 y + x 2 y 2 + 5 x 2 z 2 − 5 z 2 y 2 + 4 xy 2 z − x 2 y z + 2 xy z 2 . T 10 =     1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1     :     0 1 0 0 0 − 1 0 1 1 0 1 0 0 0 − 1 − 1     :     1 1 1 1 − 1 0 0 − 1 0 − 1 0 0 − 1 1 − 1 1     , f T 10 ( x, y , z ) = x 4 + y 4 + 2 z 4 − x 3 y + 2 x 3 z + xy 3 + 2 xz 3 − z 3 y − x 2 y 2 + 4 x 2 z 2 − 2 xy 2 z − 2 x 2 y z − xy z 2 . T 20 =     1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1     :     − 1 0 1 − 1 0 0 0 1 1 − 1 0 0 1 0 0 0     :     1 0 1 1 1 1 1 − 1 0 0 0 1 1 − 1 − 1 − 1     , 6 f T 20 ( x, y , z ) = x 4 + y 4 + 2 z 4 − x 3 y + x 3 z + y 3 z − xz 3 + 2 z 3 y + − x 2 z 2 + 6 z 2 y 2 + xy 2 z + x 2 y z + 2 xy z 2 . T 99 =     1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1     :     0 − 1 − 1 − 1 1 1 1 − 1 1 − 1 1 0 − 1 1 − 1 1     :     0 1 0 0 0 − 1 1 1 1 0 1 − 1 1 0 1 1     , f T 99 ( x, y , z ) = x 4 + 2 y 4 + 2 z 4 + 3 x 3 y + x 3 z + 3 xy 3 + y 3 z − 3 z 3 y + 6 x 2 y 2 + z 2 y 2 + 8 xy 2 z + x 2 y z − 5 xy z 2 T 119 =     1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1     :     0 0 0 − 1 − 1 1 1 − 1 0 − 1 0 0 1 0 0 0     :     1 − 1 − 1 1 1 − 1 1 0 0 1 1 1 1 1 0 − 1     , f T 119 ( x, y , z ) = x 4 + y 4 + 9 z 4 + x 3 y + xy 3 + 2 y 3 z − 2 z 3 y + 2 x 2 y 2 − 3 x 2 z 2 − z 2 y 2 + xy 2 z + x 2 y z + xy z 2 . T 207 =     1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1     :     − 1 − 1 − 1 1 0 1 1 0 − 1 0 0 − 1 − 1 1 0 − 1     :     − 1 1 − 1 1 − 1 0 1 1 − 1 0 − 1 1 0 0 − 1 − 1     , f T 207 ( x, y , z ) = x 4 + 2 y 4 + 2 z 4 − x 3 y − 3 x 3 z + xy 3 + 6 y 3 z − 3 xz 3 + 2 z 3 y − x 2 y 2 + 4 x 2 z 2 + 6 z 2 y 2 + 8 xy 2 z − 3 x 2 y z + 7 xy z 2 . T 237 =     1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1     :     0 − 1 1 − 1 0 − 1 − 1 0 0 1 − 1 0 1 1 0 1     :     0 − 1 − 1 − 1 − 1 − 1 − 1 0 0 − 1 0 1 1 − 1 1 1     , f T 237 ( x, y , z ) = x 4 + 2 y 4 + 3 z 4 − x 3 y + 4 y 3 z + 3 z 3 y + x 2 y 2 − x 2 z 2 + 6 z 2 y 2 + x 2 y z . 3 Notations from differen tial geometry F or the use in the follo wing sections, w e review some basic notations of differential geometry . F or more details, for example, see the Nomizu and Sasaki [19]. When w e denote the parametrized p oin t on the surface by x ( s, t ) , we denote its partial deriv atives by x s ( s, t ) = ∂ x ( s, t ) ∂ s , (3.1) x t ( s, t ) = ∂ x ( s, t ) ∂ t . (3.2) 7 Definition 3.3 E = h x s , x s i , F = h x s , x t i , G = h x t , x t i , (3.4) ar e c al le d the first fundamental c o efficients. Putting dx = x s ( s, t ) ds + x t ( s, t ) dt , the form I = h dx, dx i = E s 2 + 2 M dsdt + N dt 2 (3.5) is c al le d the first fundamental form of the surfac e. Definition 3.6 n = x s × x t || x s × x t || (3.7) is called the unit normal v ector at the p oin t x ( s, t ) . Definition 3.8 Putting x ss = ∂ 2 x ( s, t ) ∂ s 2 , x ss = ∂ 2 x ( s, t ) ∂ st , x ss = ∂ 2 x ( s, t ) ∂ t 2 , (3.9) the sc alar functions L = h x ss , n i M = h x st , n i N = h x tt , n i (3.10) ar e c al le d the se c ond fundamental c o efficients. The form I I = −h dX, d n i = Lds 2 + 2 M dsdt + N dt 2 (3.11) is c al le d the se c ond fundamental form of the surfac e. Definition 3.12 At the p oint P on the surfac e, let k 1 and k 2 b e the maximum and minimum of curvatur es of curves gener ate d by the interse ction of the surfac e with the plane sp anne d by the normal ve ctor and a tangent ve ctor, H = ( k 1 + k 2 ) / 2 is c al le d the me an curvatur e and K = k 1 k 2 is c al le d the Gaussian curvatur e. These ar e c alculate d by H = E N − 2 F M + GL 2( E G − F 2 ) and K = LN − M 2 E G − F 2 . (3.13) 4 Rough S L (3) in v arian ts F or c hecking GL (3)-equiv alence b et w een tw o tensors T 1 and T 2 , we need to test the equation f T 2 ( x ) = cf T 1 ( x R ) , c ∈ R and R ∈ GL (3) . F urther, S = R / | R | 1 / 3 ∈ S L (3) , cf T 1 ( x R ) = c | R | 4 / 3 f T 1 ( x R/ | R | 1 / 3 ) = c 0 f T 1 ( x S ) w ith c 0 ∈ R and S ∈ S L (3) . (4.1) Th us, GL (3)-equiv alence reduces to S L (3)-equiv alence. Then, the following theorem holds. 8 Theorem 4.2 F or two tensors T 1 and T 2 , assume that f T 2 ( x ) = cf T 1 ( x S ) do es not hold for any choic e of c ∈ R and any S ∈ S L (3) . Then, T 1 and T 2 ar e not GL (3) e quivalent. This justifies to study S L (3)-equiv alence among absolutely nonsingular tensors for in vestigating GL (3) equiv alence. Remark 4.3 If c is ne gative, then by c onsider T 3 = P T 2 with | P | < 0 , we have f T 3 ( x ) = cf T 1 ( x R ) , c ∈ R + and R ∈ GL (3) . wher e T 3 is e quivalent to T 2 . Thus, we c an assume by writing T 3 as T 2 again f T 2 ( x ) = cf T 1 ( x R ) , c ∈ R + and R ∈ GL (3) . The follo wing rough S L (3) in v arian ts are useful. Theorem 4.4 A c onvex surfac e is tr ansforme d into a c onvex surfac e by a S L (3) line ar tr ansformation and so, a tensor with a determinant p olynomial whose c onstant surfac e is c onvex is not e quivalent to a tensor with a determinant p olynomial whose c onstant surfac e is not c onvex. Only the tensor of No.1 has the conv ex surface among 8 figures in the Figure 1.1 and so the tensor of No. 1 is not S L (3) equiv alen t to all other tensors in the Figure 1. Definition 4.5 A p oint on the surfac e is c al le d a singular p oint if the normal ve ctor at the p oint c an not b e define d. Theorem 4.6 If the c onstant surfac e of a tensor T 1 has a singular p oint and the c onstant surfac e of a tensor T 2 has no singular p oint, they ar e not S L (3) ( GL (3)) e quivalent. Example 4.7 L et T 1 =     0 1 0 0 − 1 0 0 0 0 0 0 − 1 0 0 1 0     ;     0 0 1 0 0 0 0 1 − 1 0 0 0 0 − 1 0 0     ;     0 0 0 1 0 0 − 1 0 0 1 0 0 − 1 0 0 0     and T 2 =     0 1 0 0 − 1 0 0 0 0 0 1 0 0 0 0 1     ;     0 0 0 − 1 0 0 1 0 1 0 0 0 0 1 0 0     ;     0 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0     , and the determinant p olynomials of T 1 and T 2 ar e given b elow r esp e ctively. Then, we have f T 1 ( x, y , z ) = ( x 2 + y 2 + z 2 ) 2 , and f T 2 ( x, y , z ) = ( x 2 + y 2 ) 2 + z 4 . 9 Note that b oth of them ar e p ositive definite, that is, b oth of T 1 and T 2 ar e absolutely nonsingular. It is cle ar that the c onstant surfac e of f T 1 ( x ) is a spher e and it has no singular p oint, and on the other hand, that the c onstant surfac e of f T 2 ( x ) is a c onic and it has a singular p oint. Henc e, T 1 and T 2 ar e not e quivalent. Definition 4.8 When we c onsider a mesh of the p ar ameter sp ac e,it pr o duc es a lat- tic e of p oints on the c onstant surfac e. L et K + b e the numb er of lattic e p oints at which the Gaussian curvatur e is p ositive, and K − and K 0 b e define d in the same way. Then w e hav e Theorem 4.9 The triplet ( K + , K − , K 0 ) is an S L (3) -invariant. 5 S L (3) in tegral in v arian ts The follo wing Figure 2 and 3 sho ws the figures of con vex bo dies that are enclosed b y constan t surfaces. The num ber of figures corresp onds to that in our list of absolutely nonsingular tensors( Maehara [18]). In this section, w e w an t to find some S L (3)- Figure 2: The constan t surfaces for N o 1 , 19 , 22 , 23 , 42 , 60 , 61 , 65 absolutely nonsin- gular tensors in v arian t for suc h con vex b odies. F or this purp ose, the follo wing v aluation theory is a quite useful. Here, we mak e a brief summary of the v aluation theory from Ludwig [15] and [17]. The definition is stated for a general p . Definition 5.1 L et K denote the set of al l c onvex b o dies in R p . A functional t ( · ) fr om K to R is c al le d a valuation if it satisfies t ( K ) + t ( L ) = t ( K ∪ L ) + t ( K ∩ L ) , K , L ∈ K . (5.2) Next theorem is a starting p oin t of c haracterization of in v ariant v aluation. 10 Figure 3: The constant surfaces for N o 72 , 74 , 76 , 83 , 85 , 87 , 95 , 103 absolutely non- singular tensors Theorem 5.3 (Hadwiger[9]). A c ontinuous valuation t ( · ) fr om K to R is invariant with r esp e ct to rigid motion if and only if ther e ar e c onstants c 0 , c 1 , ..., c p such that t ( K ) = c 0 V 0 ( K ) + c 1 V 1 ( K ) + · · · + c p V p ( K ) , (5.4) wher e V 0 ( K ) , V 1 ( K ) , · · · , and V p ( K ) ar e the intrinsic volumes of K . We r emark that the volumes V k ar e c al le d quermassinte gr als of K in [13] and that V 0 is the Euler index, V p − 1 the affine volume of ∂ K , and V p is the volume of the c onvex b o dy K. In the fol lowing, we simply denote by a ( K ) as the affine volume and c al l it affine surfac e ar e a fol lowing [17], and deno e by V ( K ) the volume V p ( K ) . Definition 5.5 A functional t ( · ) on K is said to b e e qui-affine invariant if it is S L ( p ) -invariant and lo c ation invariant. The follo wing is essen tial for us. Theorem 5.6 (Ludwig [14] ,[16], [15] and [17] ). An upp er semi-c ontinuous val- uation t ( K ) fr om K to R is e qui-affine invariant if and only if ther e ar e c onstants c 0 , c 1 , ∈ R and c 2 ≥ 0 such that t ( K ) = c 0 V 0 ( K ) + c 1 V p ( K ) + c 2 a ( K ) , (5.7) wher e a ( K ) denotes the affine surfac e ar e a. In short, S L ( p ) in v arian t v aluation is only the weigh ted sum of the Euler index and the v olume V p ( K )and the affine surface area a ( K ). This means that Prop osition 5.8 V ( K ) and a ( K ) ar e S L -invariants. So, w e adopt the v olume V ( K ) and the affine surface area a ( K ) as indexes of S L ( p )- equiv alence. F urther, the next proposition b y Lut w ak[20] is v ery useful for us. Prop osition 5.9 When p = 3 , a ( K ) is homo gene ous of de gr e e 3 / 2 , that is, a ( dK ) = d 3 / 2 a ( K ) , K ∈ K 0 . (5.10) 11 5.1 V olume as an S L (3) -in v arian t W e are considering the equiv alence relation among absolutely nonsingular tensors. As is shown in Theorem 2.7, for such kind of tensors, the constan t surfaces of them are compact. Note that from Propisition 5.8 the volume of the region enclosed by the constan t surface is S L (3)-in v arian t. Then, by the follo wing Gauss’s theorem, w e can calculate the v olume b y the parametric represen tation given b y the equation (2.6). Theorem 5.11 (Gaussian formula) F or the r e gion Ω enclose d by the sp ac e surfac e ∂ Ω , letting f dy ∧ dz + g dz ∧ dx + hdx ∧ dy b e the differ ential form of 2nd de gr e e, it holds Z ∂ Ω f dy ∧ dz + g dz ∧ dx + hdx ∧ dy = Z Ω  ∂ f ∂ x + ∂ g ∂ y + ∂ h ∂ z  dx ∧ dy ∧ dz (5.12) W e denote b y V (Ω) the volume of the region Ω. By this form ula, we hav e V (Ω) = LH S of the eq uation (5 . 12) . (5.13) F or the present case, b y use of the spherical co ordinates (s,t), the p oin t of the b oundary ∂ Ω is parametrized as x = r ( s, t )(Φ x ( s, t ) , Φ y ( s, t ) , Φ z ( s, t )) . Hence, w e ha ve dy = ∂ y ∂ s ds + ∂ y ∂ t dt (5.14) =  ( − 1 / 4) dp/ds p 5 / 4 Φ y + d Φ y /ds p 1 / 4  ds +  ( − 1 / 4) dp/dt p 5 / 4 Φ y + Φ y /dt p 1 / 4  dt dz = ∂ z ∂ s ds + ∂ z ∂ t dt, =  ( − 1 / 4) dp/ds p 5 / 4 Φ z + d Φ z /ds p 1 / 4  ds +  ( − 1 / 4) dp/dt p 5 / 4 Φ z + Φ z /dt p 1 / 4  dt. (5.15) Therefore, dy ∧ dz =  ( − 1 / 4) dp/ds p 5 / 4 Φ y + d Φ y /ds p 1 / 4   ( − 1 / 4) dp/dt p 5 / 4 Φ z + Φ z /dt p 1 / 4  ds ∧ dt −  ( − 1 / 4) dp/dt p 5 / 4 Φ y + Φ y /dt p 1 / 4   ( − 1 / 4) dp/ds p 5 / 4 Φ z + d Φ z /ds p 1 / 4  ds ∧ dt. Similarly , dz ∧ dx =  ( − 1 / 4) dp/ds p 5 / 4 Φ z + d Φ z /ds p 1 / 4   ( − 1 / 4) dp/dt p 5 / 4 Φ x + Φ x /dt p 1 / 4  ds ∧ dt −  ( − 1 / 4) dp/dt p 5 / 4 Φ z + Φ z /dt p 1 / 4   ( − 1 / 4) dp/ds p 5 / 4 Φ x + d Φ x /ds p 1 / 4  ds ∧ dt, 12 and dx ∧ dy =  ( − 1 / 4) dp/ds p 5 / 4 Φ x + d Φ x /ds p 1 / 4   ( − 1 / 4) dp/dt p 5 / 4 Φ y + Φ y /dt p 1 / 4  ds ∧ dt −  ( − 1 / 4) dp/dt p 5 / 4 Φ x + Φ x /dt p 1 / 4   ( − 1 / 4) dp/ds p 5 / 4 Φ y + d Φ y /ds p 1 / 4  ds ∧ dt. By using these, w e can calculate the v olume of the region enclosed by the constan t surface of the determinan t p olynomial. Let T 1 and T 2 b e t wo n × n × 3 tensors and let Ω i denote the regions { bmx | f T i ( x ) ≤ 1 } , whic h are enclosed b y the surfaces of { x | f T 1 ( x ) = 1 } . Then, by S L (3) in v ariance of v olumes, w e ha v e Theorem 5.16 If V 1 6 = V 2 , f T 2 ( x ) 6 = f T 1 ( x R ) for any R ∈ S L (3) , namely, T 1 and T 2 ar e not S L ( n ) × S L ( n ) × S L (3) e quivalent. F or GL in v ariance, the next lemma is helpful. Lemma 5.17 F or a determinant p olynomial f ( x ) , let V ( c ) b e the volume of Ω c = { x | cf ( x ) ≤ 1 } . Then V ( c ) = c − 3 / 4 V (1) for 4 × 4 × 3 c ase. Pro of By c hanging a p olynomial f ( x ) into its constan t multiple cf ( x ), the co or- dinates x ( s, t ) on the constan t surface are sub ject to c hanges to ( 1 c ) 1 / 4 x ( s, t ). Hence, the in tegral 1 3 Z ∂ Ω xdy ∧ dz + y dz ∧ dx + z dx ∧ dy (5.18) is m ultiplied by c − 3 / 4 . This prov es the assertion of Lemma 5.17. Theorem 5.19 Assume that T 1 and T 2 b e S L (3) e quivalent and ther efor e that ther e is a r elation b etwe en their determinant p olynomials, f T 2 ( x ) = cf T 1 ( x R ) , (5.20) wher e c ∈ R and R ∈ S L (3) . L et V 1 ( c ) and V 2 ( c ) denote the volumes of Ω (1) c = { x | cf T 1 ( x ) ≤ 1 } and Ω (2) c = { x | cf T 2 ( x ) ≤ 1 } r esp e ctively. Then, it holds that c = ( V 1 /V 2 ) 4 / 3 . (5.21) Pro of The pro of is trivial from Lemma 5.17 and omitted. F rom Theorem 5.19, w e can know the constan t c in the equation (5.20). In the next section, it will be made clear that this expresson is helpful for establishing GL (3)-equiv alence. 13 5.2 Affine surface area as an S L (3) -in v arian t In this section, for testing S L (3)-equiv alence, w e prop ose to use the affine surface area, whic h is an S L (3)-in v arian t by Theorem 5.8. When p = 3, the affine surface area has the follo wing in tegral expression. Definition 5.22 F or a smo oth c onvex b o dy K ⊂ R 3 , the affine surfac e ar e a is given by a ( K ) = Z ∂ K κ ( K, x ) 1 4 √ E G − F 2 dsdt, (5.23) wher e κ ( ∂ K , x ) is the Gaussian curvatur e and E , F and G denote the first funda- mental c o efficients. Next, we show that the affine surface area is useful ev en as a tester of GL (3)- equiv alence. Assume that we kno w the constan t c in the relation f T 2 ( x ) = cf T 1 ( x R ) with c ∈ R + and R ∈ S L (3) b y Theorem 5.19. Then, Ω 2 = { x | f T 2 ( x ) ≤ 1 } (5.24) = { x | cf T 1 ( x ) ≤ 1 } = { x | f T 1 ( c 1 / 4 x ) ≤ 1 } = c − 1 / 4 { x | f T 1 ( x ) ≤ 1 } = c − 1 / 4 Ω 1 . (5.25) F rom Prop osition 5.9, we hav e a (Ω 2 ) = c − 3 / 8 a (Ω 1 ) . (5.26) Th us, w e ha v e the following. Theorem 5.27 L et T 1 and T 2 b e absolutely nonsingular tensors. Noting R emark 4.3, by The or em 5.19, we c an obtain the estimate of c ∈ R + under the assumption that their determinant p olynomials have the r elation f T 2 ( x ) = cf T 1 ( x R ) for some unknown c onstant c ∈ R + and an unknown matrix R ∈ S L (3) . Then, if a (Ω 2 ) 6 = c − 1 / 8 a (Ω 1 ) , T 1 and T 2 ar e not GL (3) e quivalent. By using Theorem 5.17, this is rephrased as Theorem 5.28 L et T 1 and T 2 b e absolutely nonsingular tensors. Then, if a (Ω 2 ) 6 =  V (Ω 2 ) V (Ω 1 )  1 / 2 a (Ω 1 ) , (5.29) T 1 and T 2 ar e not GL (3) -e quivalent, wher e V (Ω 1 ) and V (Ω 2 ) denote the volume of Ω 1 and Ω 2 r esp e ctively. 14 6 In tegral GL (3) -in v arian t In the latter half of the previous section, w e presen ted a pro cedure to test a non- GL (3)-equiv alence, ho w ever, it is somewhat indirect b ecause we need to estimate the constan t c before starting the procedure. In this section, we consider a direct metho d handling non-equiv alence b y using a generalized affine surface area. That is, we consider the L q affine surface area, whic h is an extension of the affine surface area and dev elop ed b y Letw ak [20]. Hug [10] ga ve an equiv alen t definition. The follo wing is the Hug’ s definition. Definition 6.1 L q a ( K ) = Z ∂ K κ 0 ( K, x ) q p + q dσ K ( x ) (6.2) wher e κ 0 ( K, x ) = κ ( K, x ) h x , n ( K , x ) i p +1 . (6.3) and dσ K ( x ) is c al le d a c one me asur e define d by dσ K ( x ) = h x , n ( K , x ) i d x , (6.4) and n ( K , x ) denotes the outer normal at x on ∂ K . When q = 1, L q ( K ) b ecomes the affine surface area a ( K ), and when q = p , it b ecomes a classical centro-affine surface area a c ( K ) th ta is defined as a c ( K ) = Z ∂ K κ 0 ( K, x ) 1 / 2 dσ K ( x ) , (6.5) whic h is known to be GL ( p )-inv ariant. The c haracterization of a general GL ( p )- in v arian t functional is giv en below. Theorem 6.6 ( Ludwig and Reitzsner [17] ) L et K 0 b e the sp ac e of c onvex b o dies that c ontain the origin in their interiors. A n upp er semi-c ontinuous functional t ( · ) fr om K 0 to R 1 is GL ( p ) -invariant if and only if ther e ar e nonengative c onstants c 0 and c 1 such that t ( K ) = c 0 V 0 ( K ) + c 1 a c ( K ) . (6.7) 7 Spherical design According to our exp erimen ts, the n umerical integrations of the in v arian ts m ust b e accurate at least 2 decimals. So, the caluculations of the in v arian ts are a little bit heavy . In this section, we consider the t-design metho d as an substitute of the nemerical in tegrations. The spherical design was initiated by Delsarte et al. [7] and has b een studied by sev eral researc hers, for example, see Bannai and Bannai [3]. It is defined as follo ws. 15 7.1 An o v erview of spherical design Definition 7.1 A finite set X on the spher e is c al le d t-spheric al design if the fol- lowing e quality holds that for any p olynomial f ( x, y , z ) with a de gr e e less than or e qual to t, 1 | S 2 | Z S 2 f ( x, y , z ) dσ = 1 | X | X ( x,y ,z ) ∈ X f ( x, y , z ) , (7.2) wher e S 2 denotes the unit spher e of R 3 and dσ denotes the surfac e element of the spher e and | S 2 | denotes the surfac e ar e a of the spher e. A parametrized integral form ula of the equation 7.2 is giv en b y Z f ( s, t ) sin( s ) dsdt = 4 π N N X i =1 f ( s i , t i ) , (7.3) where ( s i , t i ) , i = 1 , 2 , ...., N are the corresp onding parameters to the design p oin ts in X . One p oin t to ov ercome for our purp ose is that w e need to in tegrate some nonlinear functions that are not polynomials and hence w e can not use an y t-design directly . How ev er, w e can rely on the next theorem to solv e this p oin t. Theorem 7.4 L et f ( x, y , z ) b e an c ontinuous function over the unit spher e and let  t b e a p ositive numb er such that | f ( x, y , z ) − p ( x, y , z ) | <  t uniformly for some p olynomial p ( x, y , z ) with de gr e e less than or e qual to t . Then, it holds that | Z ∂ S f ( x, y , z ) dS − 4 π 1 N N X i =1 f ( x i ) | < 8 π  t (7.5) Pro of      Z ∂ S f ( x, y , z ) dS − 4 π 1 N N X i =1 f ( x i )      (7.6) ≤      Z ∂ S f ( x, y , z ) dS − 4 π 1 N N X i =1 p ( x i )      +      4 π 1 N N X i =1 p ( x i ) − 4 π 1 N N X i =1 f ( x i )      =     Z ∂ S f ( x, y , z ) dS − Z | ∂ S p ( x i ) dS     + 4 π N      N X i =1 | p ( x i ) − f ( x i ) |      ≤ Z  t dS + 4 π  t = 8 π  t 16 Remark 7.7 By the ab ove the or em, we ne e d not to know the b est appr oximate p oly- nomial c oncr etely in or der to obtain an appr oximate value of the inte gr ation, and it is enough to use f ( x, y , z ) itself. Mor e over the err or of the appr oximation is b ounde d fr om ab ove by the multiple of  t by 8 π . F or a substantial evaluation of the appr ox- imation, we ne e d to know  t . The pr oblem is inter esting, however, it is a little bit he avy task at pr esent, and so it is p ostp one d to the futur e work. 7.2 Calculation of in tegral in v arian ts b y a 20-design Using the result of the previous subsection, we consider the integration a ( K ) = Z κ ( s, t ) 1 / 4 √ E G − F 2 dsdt. (7.8) where s, t mo ves 0 < s < π ,0 < t < 2 π . This in tegration can b e thought to b e an in tegration o v er the unit sphere b y a ( K ) = Z ( s,t, ) ∈ [0 ,π ] × [0 , 2 π ] κ ( s, t ) 1 / 4 √ E G − F 2 dsdt (7.9) = Z ( s,t, ) ∈ [0 ,π ] × [0 , 2 π ] κ ( s, t ) 1 / 4 √ E G − F 2 sin s sin sdsdt = Z ∂ S κ ( s, t ) 1 / 4 √ E G − F 2 sin s dS, (7.10) where dS = sin ( s ). Hence, p ( x, y , z ) = κ ( s, t ) 1 / 4 √ E G − F 2 sin s (7.11) is taken to b e a function ov er the unit sphere and so the integral in v arian t can b e appro ximated b y the right hand side of the equation b elo w. Z ∂ S κ ( s, t ) 1 / 4 √ E G − F 2 sin s dS ∼ 4 π N N X i =1 p ( x i , y i , z i ) (7.12) The v alues of inv arian ts calculated b y the lattice metho d and the 20-design metho d will be giv e in the next section. The 20-design metho d sho w v ery nice approxi- mations in some cases, how ev er, do not show goo d appro ximations for other cases. That is, for our in tegration of inv ariants, the spherical design metho d do es not give stable v alues, unfortunately . This migh t suggest that we need to use design with more higher degree than 20. 8 Effectiv eness of the in v arian ts as testers of non- equiv alence In this section, w e will show the effectiveness of the numerical v alues of the inv arian ts as testers of non-equiv alence. W e n umerically calculated the v olume V (Ω), the affine 17 surface area a (Ω)and cen tro-affine surface area a c (Ω) of the region Ω = { x | f ( x ≤ 1 } defined by the determinant p olynomials f T ( x ). As examples, w e calculate them for the 16 tensors whic h are in K 0 , whose constant surfaces are figured in Figures 2 and 3 in the section 5. The n umerical calculations are performed in tw o wa y , that is, b y the lattice metho d and b y the t-design metho d, and they are compared. As for the t-design method, w e use the 20-design named des.3.216.20 in [8] whic h has 216 p oin ts. In the tables below, M1-P2-G5, M6-P2-G, M1-P2-G7 and 20-design denote the globally adaptive in tegration with accuracy of 5 digits, pseudo-Monte Carlo in tegration, the globally adaptiv e integration with accuracy of 7 digits by 64 decimal calculation and 20-design metho d by IEEE754 decimal calculation, respectively . F or all calculation w ere done by Mathematica. T able 1 sho ws that the SL inv ariance of T ensor V0 V1 V2 V3 T001 2.9197794095194 2.9197794099529 2.9197794089308 2.9197794061274 T019 4.0314824331814 4.0314824340674 4.0314824332515 4.0314824319603 T022 3.6306602017309 3.6306602004447 3.6306602054741 3.6306602016552 T023 3.4355628950802 3.4355628819358 3.4355628878857 3.4355628897838 T042 3.7515624235646 3.7515624142272 3.7515624197586 3.7515624152774 T060 2.1440485535226 2.1440485507771 2.1440485551454 2.1440485550215 T061 2.8594583429857 2.8594583441125 2.8594583445567 2.8594583445567 T065 3.1084258968340 3.1084258946417 3.1084258957994 3.1084258984271 T072 4.6861403575076 4.6861403597489 4.6861403542060 4.6861403560079 T074 3.6302252513670 3.6302253269919 3.63022533280632 3.6302253350968 T able 1: V olumes b y M1-P2-G7 : T n , where n = 001 , 0019 , 022 , 023 , 042 , 060 , 061, 065 , 072 and 074 Each line denoted as T n-0 lists the v alue of the original tensor and the lines T n-i, i = 1 , 2 , ..., 5 list the v alues for the transformed tensors of Tn b y a randomly c hosen matrix of S L (3). v olumes of the redions enclosed by the constan t surface is clealry seen numerically for ev ery absolutely nonsingular chosen tensors. T ables 2 and 3 of the affine surface area sho w that the affine suraface area is SL in v arian t and that all relev ant tensors are not S L (4) × S L (4) × S L (3) equiv alen t m utually . F rom Theorem 5.28, combining the volume data, w e also conclude that they are not GL equiv alen t. This last fact is also deriv ed by a direct usage of the cen tro-affine surface data which is seen in T able 4 and T able 5. Indeed, T ables 4 and 5 show that the cen tro-affine surface area is really GL in v arinat, and that three point decimal accuracy will b e sufficient to detect non GL (3)-equiv alence betw een 4 × 4 × 3 absolutely nonsingular tensors, whose elements consists of only -1,0,1. The M1-P2-G7 metho d seems clearly the b est for discrimi- nating the tensors relating to GL nonequiv alence. 18 T ensor M1-P2-G5 M6-P2-G5 M1-P2-G7 20-design T001-0 9.961493457 9.962796404 9.961471493 9.90317 T001-1 9.961470358 9.961249135 9.961471489 8.73023 T001-2 9.961471133 9.971266750 9.961471486 9.96328 T001-3 9.961470327 9.959509709 9.961471474 9.79057 T001-4 9.961471186 9.979456186 9.961471478 10.88367 T001-5 9.961474220 9.997180989 9.961471490 10.99278 T019-0 11.560007113 11.560055546 11.560007991 11.87277 T019-1 11.560007742 11.552017302 11.560007993 11.69239 T019-2 11.560008558 11.559866424 11.560007993 11.51344 T019-3 11.560007692 11.501609971 11.560007989 13.36769 T019-4 11.560008494 11.545260017 11.560007991 10.40203 T019-5 11.560001924 11.558176522 11.560007996 11.49393 T022-0 11.020675684 11.024551464 11.020674135 11.05202 T022-1 11.020673831 11.016947424 11.020674138 11.45345 T022-2 11.020676195 11.027525350 11.020674147 11.54386 T022-3 11.020673214 11.016006939 11.020674140 11.07524 T022-4 11.020675399 11.031596952 11.020674133 11.37096 T022-5 11.020674431 11.022431931 11.020674135 10.74089 T023-0 10.771760482 10.773095422 10.771760351 10.73293 T023-1 10.771758801 10.774759865 10.771760349 9.26881 T023-2 10.771759301 10.725843291 10.771760352 10.94587 T023-3 10.771759730 10.766135806 10.771760352 13.23848 T023-4 10.771757516 10.773059224 10.771760350 10.78732 T023-5 10.771759533 10.773749461 10.771760351 10.88149 T042-0 11.136697741 11.136755128 11.136697332 10.99424 T042-1 11.136695637 11.140565725 11.136697314 11.28015 T042-2 11.136699257 11.140835239 11.136697323 11.89052 T042-3 11.136721676 11.203272583 11.136697308 12.13058 T042-4 11.136696147 11.106329119 11.136697270 9.81007 T042-5 11.136697731 11.107102150 11.136697313 13.99432 T able 2: Affine surface area: T n , where n = 001 , 019 , 022 , 023 and 042 Eac h line denoted as T n-0 lists the v alue of the original tensor and the lines T n-i, i = 1 , 2 , ..., 5 list the v alues for the transformed tensors of Tn b y a randomly c hosen matrix of S L (3). 19 T ensor M1-P2-G5 M6-P2-G5 M1-P2-G7 20-design T060-0 8.704587985 8.705126156 8.704588101 8.74300 T060-1 8.704590255 8.781085267 8.704588109 8.50058 T060-2 8.704596276 8.705498658 8.704588101 8.73210 T060-3 8.704588380 8.711001740 8.704588104 8.80910 T060-4 8.704586669 8.703029024 8.704587984 8.85973 T060-5 8.704588143 8.705901809 8.704588100 8.56701 T061-0 9.759043314 9.759635865 9.759045706 9.741275 T061-1 9.759036154 9.759076500 9.759045704 9.72403 T061-2 9.759050068 9.748685352 9.759045707 9.56041 T061-3 9.759044653 9.734392909 9.759045710 9.76040 T061-4 9.759058206 9.745677922 9.759045694 10.37056 T061-5 9.759046974 9.755984062 9.759045716 8.72501 T065-0 10.273389075 10.274251947 10.273389369 10.33927 T065-1 10.273387633 10.260497042 10.273389360 10.76367 T065-2 10.273389342 10.277789370 10.273389368 10.26620 T065-3 10.273388029 10.249526640 10.273389366 10.42661 T065-4 10.273389939 10.276245030 10.273389370 10.06295 T065-5 10.273397527 10.279052599 10.273389365 10.33636 T072-0 12.483701912 12.483843205 12.483691274 12.67586 T072-1 12.483689616 12.484388161 12.483691282 12.38965 T072-2 12.483690234 12.481034408 12.483691282 10.30731 T072-3 12.483690116 12.498107747 12.483691264 11.96858 T072-4 12.483698348 12.435166726 12.483691276 11.219584 T072-5 12.483686195 12.508162438 12.483691276 10.183837 T074-0 10.732327625 10.732623087 10.732332110 10.80078 T074-1 10.732332283 10.726775221 10.732332117 10.55820 T074-2 10.732337739 10.734008328 10.732332113 11.20578 T074-3 10.732332889 10.724453313 10.732332112 10.87848 T074-4 10.732331733 10.729148909 10.732332214 10.95073 T074-5 10.732329467 10.727707310 10.732332111 10.57017 T able 3: Affine surface area: T n , where n = 060 , 061 , 065 , 072 and 074 Eac h line denoted as T n-0 lists the v alue of the original tensor and the lines T n-i, i = 1 , 2 , ..., 5 list the v alues for the transformed tensors of Tn b y a randomly c hosen matrix of S L (3). 20 T ensor M1-P2-G7 M6-P2-G5 M1-P2-G5 20-design T001-0 11.690150892617500 11.687899476332365 11.687898336789288 11.59968 T001-1 11.751922920525157 11.687898955213611 11.687898343722015 8.421025 T001-2 11.689694693901319 11.687898370365255 11.687898355824357 11.68469 T001-3 11.721355953709315 11.687894829195568 11.687898343333765 11.29880 T001-4 11.692877418652227 11.687897242831659 11.687897875631138 10.59083 T001-5 11.679997753276740 11.687897167430656 11.687898359334835 11.46900 T019-0 11.509733354093680 11.509334536488204 11.509333804897551 11.81248 T019-1 11.472821548199051 11.509332873485376 11.509333807975230 12.65290 T019-2 11.509963231209824 11.509337447098934 11.509333799194381 11.32764 T019-3 11.552527050941017 11.509334159193976 11.509333800434498 11.38444 T019-4 11.495864684062547 11.509335287391230 11.509333801132503 11.37034 T019-5 11.522546264759133 11.509333512497239 11.509333798526679 22.50564 T022-0 11.574282949497377 11.568790730790213 11.568790156808308 11.59771 T022-1 11.570887655990263 11.568785645774059 11.568790144452251 11.63356 T022-2 11.568787229271756 11.568788230429048 11.568790347476747 11.90185 T022-3 11.567926875696718 11.568789765657722 11.568790134358534 11.57677 T022-4 11.597381830882211 11.568789372237021 11.568790132199215 13.39389 T022-5 11.561310960443544 11.568790290463560 11.568790451975676 11.44911 T023-0 11.631078897689606 11.626439742081966 11.626439153758515 11.57877 T023-1 11.619997976153462 11.626439518693340 11.626439146934238 11.39835 T023-2 11.611266008293132 11.626440231360507 11.626439154231153 11.43501 T023-3 11.647477652583963 11.626433368429471 11.626439151521914 13.20628 T023-4 11.607565993095791 11.626439544654837 11.626439154062155 10.72795 T023-5 11.620421522261536 11.626439658214246 11.626439152653352 11.74869 T042-0 11.502624357421948 11.504755263366923 11.504752079092657 11.30545 T042-1 11.507105268508006 11.504753311150279 11.504752086650519 11.07124 T042-2 11.519424921951189 11.504753899401661 11.504752079220612 9.41924 T042-3 11.501095106227783 11.504764044950799 11.504752085646500 12.16150 T042-4 11.530791206130419 11.504752412140897 11.504752073761618 10.11515 T042-5 11.499503647742464 11.504752956532382 11.504752076792938 10.64605 T able 4: Cen tro-affine surface area: T n , where n = 001 , 019 , 022 , 023 and 042 Eac h line denoted as T n-0 lists the v alue of the original tensor and the lines T n-i, i = 1 , 2 , ..., 5 list the v alues for the transformed tensors of Tn by a randomly chosen matrix of GL (3). 21 T ensor M6-P2-G5 M1-P2-G5 M1-P2-G7 20-design T060-0 11.989971644000403 11.989476119401702 11.989477685702977 12.05017 T060-1 11.990418566344240 11.989479611584825 11.989477723348062 12.01483 T060-2 11.976852295964006 11.989478648296963 11.989477738864300 11.80414 T060-3 11.994778478025750 11.989477993940143 11.989477740049253 11.64148 T060-4 11.989039776987073 11.989478217916079 11.989477724638414 12.09719 T060-5 12.046738095770048 11.989477160425962 11.989477721024015 12.11083 T061-0 11.519673795399891 11.518135424201142 11.518135117247486 11.48023 T061-1 11.518661618867961 11.518134427948641 11.518135113069415 11.45852 T061-2 11.519323023325257 11.518135433182912 11.518135109619174 11.45517 T061-3 11.517878708284445 11.518130456996306 11.518135109891727 11.66299 T061-4 11.517606307852915 11.518129721219993 11.518135107795112 11.47192 T061-5 11.531441863084249 11.518134741001642 11.518135110921667 10.31802 T065-0 11.660951650838694 11.660077139783777 11.660146606151409 11.77330 T065-1 11.657097464096733 11.660146835636129 11.660146583158155 11.64542 T065-2 11.662671583310311 11.660135616613492 11.660146602669996 11.69570 T065-3 11.657074111599187 11.660148534680922 11.660146593520388 11.58150 T065-4 11.661605706427124 11.660146860738219 11.660146601870989 12.45337 T065-5 11.668831112582931 11.660147254608734 11.660146596730511 11.76209 T072-0 11.545589518947570 11.545142097769179 11.545141226929544 11.76647 T072-1 11.545716622630585 11.545139592226001 11.545141221483210 11.52675 T072-2 11.562200209044268 11.545142319259361 11.545141224116661 8.50885 T072-3 11.575704218963165 11.545144648175810 11.545141226744587 10.30769 T072-4 11.535076590703719 11.545140326506765 11.545141235723655 12.22675 T072-5 11.545910129113601 11.545141536682674 11.545141208615009 11.19618 T074-0 11.116314213623787 11.116088526600165 11.116090556639371 11.20632 T074-1 11.109183432623630 11.116086365050504 11.116090553382580 11.09112 T074-2 11.121063605466493 11.116094135711593 11.116090554260284 9.58706 T074-3 11.090134159234779 11.116089713933696 11.116090550551305 10.19594 T074-4 11.135697898280837 11.116091869446610 11.116090554811293 11.15140 T074-5 11.117481923868303 11.116094503240262 11.116090554606608 11.08749 T able 5: Cen tro-affine surface area: T n , where n = 060 , 061 , 065 , 072 and 074 Eac h line denoted as T n-0 lists the v alue of the original tensor and the lines T n-i, i = 1 , 2 , ..., 5 list the v alues for the transformed tensors of Tn by a randomly chosen matrix of GL (3). 22 9 Conclusion W e treated the S L (4) × S L (4) × S L (3) , GL (4) × GL (4) × S L (3) or GL (4) × GL (4) × GL (3) non-equiv alence problem of 4 × 4 × 3 absolutely nonsingular tensors. W e prop osed a metho d to addres to the problem through the determinan t p olynomials. F urthermore we prop osed to solv e the problem b y differen tial geometric S L (3) or GL (3) inv arian t of the constan t surface of the determinant p olynomials. F rom the n umerical analysis by Mathematica, it was sho wn that the stable v alues of in v arian ts are obtainable numerically and also it was shown that the affine surface area and the cen tro-affine surface area are useful to detect the non-equiv alence. This means that the algebraic problem: whether a system of algebraic equations with many v ariables can ha v e real solutions or not, can b e resolv ed b y differential geometric metho ds. It is a nice link b et ween algebra and differen tial geometry . Second, we in vestigated the spherical design metho d for calculating inv arian ts. At present, w e think that the v alues given b y the adaptiv e lattice metho ds are more reliable than those giv en b y the spherical design metho d. 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