Narrowing the Gap: SOS Ranks of $4 \times 3$ Biquadratic Forms and a Lower Bound of $8$

Narro wing the Gap: SOS Ranks of 4 × 3 Biquadratic F orms and a Lo w er Bound of 8 Yi Xu ∗ Ch unfeng Cui † and Liqun Qi ‡ F ebruary 26, 2026 Abstract W e in vestigate the maxim um sum-of-squares (SOS) rank of biquadratic forms in the critical case of 4 × 3 v ariables, where the general bounds are currently 7 ≤ BSR(4 , 3) ≤ 11. By analyzing t wo important structured sub classes, we obtain exact determinations and impro ved upp er b ounds that significantly narro w this gap. F or simple biquadratic forms those containing only distinct terms of the t yp e x 2 i y 2 j w e pro ve that the maximum ac hiev able SOS rank is exactly 7, a v alue attained by a form corresp onding to a C 4 -free bipartite graph with the maximum num b er of edges. This settles the question for simple forms. F or y -deficient biquadratic forms a class introduced here that p ermits cross terms among tw o of the three y -v ariables while the third appears only in pure square terms we pro ve an upp er b ound of 9 by com bining Calder¨ on’s theorem on m × 2 forms with the kno wn v alue BSR(4 , 2) = 5. Our main result is a constructive pro of that BSR(4 , 3) ≥ 8. W e presen t an explicit non- simple, non-deficient 4 × 3 biquadratic form and pro v e it requires exactly eight squares, thereb y improving the general low er b ound. This sho ws that any form achieving a rank higher than 8 m ust p ossess a more complex algebraic structure, and it redu ces the search space for determining the true v alue of BSR(4 , 3). Connections to Zarankiewicz num b ers, extremal graph theory , and classical results on sums of squares are highlighted throughout. Keyw ords. Biquadratic forms, sum-of-squares, SOS rank, simple forms, y -deficien t forms, diagonal forms, p ositive semidefinite, Zarankiewicz num b er. AMS sub ject cl assifications. 11E25, 12D15, 14P10, 15A69, 90C23. ∗ Sc ho ol of Mathematics, Southeast Universit y , Nanjing 211189, China. Nanjing Center for Applied Mathe- matics, Nanjing 211135, China. Jiangsu Provincial Scien tific Research Center of Applied Mathematics, Nanjing 211189, China. ( yi.xu1983@hotmail.com ) † Sc ho ol of Mathematical Sciences, Beihang Universit y , Beijing 100191, China. ( chunfengcui@buaa.edu.cn ) ‡ Departmen t of Applied Mathematics, The Hong Kong Polytec hnic Universit y , Hung Hom, Kowloon, Hong Kong. ( maqilq@polyu.edu.hk ) 1 1 In tro duction Let m, n ≥ 2. A biquadratic form in v ariables x = ( x 1 , . . . , x m ) and y = ( y 1 , . . . , y n ) is a homogeneous polynomial P ( x , y ) = m X i,k =1 n X j,l =1 a ij k l x i x k y j y l , with real co efficien ts a ij k l . It is called p ositive semidefinite (PSD) if P ( x , y ) ≥ 0 for all x , y , and sum-of-squar es (SOS) if it can b e written as a finite sum of squares of bilinear forms. The smallest n um b er of squares required is the SOS r ank of P , denoted sos ( P ). Let B S R ( m, n ) b e the maximum SOS rank of m × n SOS biquadratic forms Curren tly , w e kno w the follo wing ab out B S R ( m, n ). • B S R ( m, 2) = m + 1, i.e., B S R (2 , n ) = n + 1 [1]. • B S R (3 , 3) = 6 [2] • B S R ( m, n ) ≤ mn − 1 [7]. • B S R ( m, n ) ≥ z ( m, n ), where z ( m, n ) is the Zarankiewicz n umber [5]. Th us, the current frontier is on BSR(4 , 3). As z (4 , 3) = 7, w e no w kno w that 7 ≤ BSR(4 , 3) ≤ 11. In this pap er, w e mak e substan tial progress in narro wing this gap. First, we conduct a systematic analysis of t wo fundamen tal sub classes. F or simple bi- quadratic forms (Section 2), we lev erage their connection to bipartite graphs and Zarankiewicz n umbers to prov e that the maxim um SOS rank is exactly 7. F or y -deficien t biquadratic forms (Section 3), a new class introduced here, we use a splitting argumen t to establish an upp er b ound of 9. Our main contribution is a constructive impro vemen t of the general lo wer b ound. In Section 4, w e present an explicit 4 × 3 biquadratic form that is neither simple nor y -deficient and prov e that its SOS rank is exactly 8 (Theorem 4.1). This result, BSR(4 , 3) ≥ 8, is the cen tral finding of our w ork. It demonstrates that the maximum rank for the general case is at least 8 and implies that an y form ac hieving a rank of 9 or higher must p ossess a more complex algebraic structure than those considered here. The remainder of this paper is organized as follows. In Section 2, we fo cus on simple biquadratic forms and prov e that the maximum SOS rank for 4 × 3 simple forms is exactly 7. In Section 3, w e in tro duce the class of y -deficient biquadratic forms, which may con tain cross terms, and sho w that their SOS rank is at most 9. Diagonal forms app ear as a sp ecial case, and w e also mention an alternativ e proof using a row split. In Section 4, we construct and analyze the form Q , pro ving it requires eight squares and establishing the new lo w er b ound BSR(4 , 3) ≥ 8. Finally , in Section 5, we conclude with a summary of our findings and a discussion of open problems and future researc h directions. 2 2 Simple Biquadratic F orms Let m ≥ n . A biquadratic form is called simple if it contains only distinct terms of the t yp e x 2 i y 2 j . W e define a family of simple forms P m,n,s where s = 1 , . . . , mn coun ts the num b er of square terms, follo wing a fixed ordering of index pairs ( i, j ) (see [7] for details). F or m = 4 and n = 3 the first sev en forms are: P 4 , 3 , 1 = x 2 1 y 2 1 , P 4 , 3 , 2 = x 2 1 y 2 1 + x 2 2 y 2 2 , P 4 , 3 , 3 = x 2 1 y 2 1 + x 2 2 y 2 2 + x 2 3 y 2 3 , P 4 , 3 , 4 = x 2 1 y 2 1 + x 2 2 y 2 2 + x 2 3 y 2 3 + x 2 1 y 2 2 , P 4 , 3 , 5 = x 2 1 y 2 1 + x 2 2 y 2 2 + x 2 3 y 2 3 + x 2 1 y 2 2 + x 2 2 y 2 3 , P 4 , 3 , 6 = x 2 1 y 2 1 + x 2 2 y 2 2 + x 2 3 y 2 3 + x 2 1 y 2 2 + x 2 2 y 2 3 + x 2 3 y 2 1 , P 4 , 3 , 7 = x 2 1 y 2 1 + x 2 2 y 2 2 + x 2 3 y 2 3 + x 2 1 y 2 2 + x 2 2 y 2 3 + x 2 3 y 2 1 + x 2 4 y 2 1 . Eac h of them corresp onds to a 4 × 3 bipartite graph G = ( S, T , E ), where the vertex sets S = { 1 , 2 , 3 , 4 } and T = { 1 , 2 , 3 } , and E is the edge set. F or P 4 , 3 , 7 , its edge set E = { (1 , 1) , (2 , 2) , (3 , 3) , (1 , 2) , (2 , 3) , (3 , 1) , (4 , 1) } . All of these sev en bipartite graphs are C 4 -free, i.e., none of them contains a C 4 cycle C ij k l ≡ { ( i, j ) , ( i, l ) , ( k , j ) , ( k , l ) } with i  = k and j  = l . By [5], the SOS rank of P 4 , 3 ,p equals the size of its edge set. In particular, as the bipartite graph corresp onding to P 4 , 3 , 7 is a 4 × 3 bipartite graph with the maximum C 4 -free edge set, b y [5] w e ha ve the following theorem. Theorem 2.1 (A 4 × 3 simple form requiring seven squares) . The form P 4 , 3 , 7 ( x , y ) = x 2 1 y 2 1 + x 2 2 y 2 2 + x 2 3 y 2 3 + x 2 1 y 2 2 + x 2 2 y 2 3 + x 2 3 y 2 1 + x 2 4 y 2 1 satisfies sos( P 4 , 3 , 7 ) = z (4 , 3) = 7 . T o b ound the SOS rank of simple forms with more terms, w e use a com binatorial observ ation. F or i  = k and j  = l define P ij k l ( x , y ) = x 2 i y 2 j + x 2 k y 2 l + x 2 i y 2 l + x 2 k y 2 j = ( x i y j + x k y l ) 2 + ( x i y l − x k y j ) 2 , so sos( P ij k l ) = 2. Lemma 2.2. Every simple biquadr atic form in variables ( x 1 , x 2 , x 3 , x 4 ) and ( y 1 , y 2 , y 3 ) c ontain- ing at le ast 8 distinct terms x 2 a y 2 b c ontains C ij k l for some i  = k , j  = l . Prop osition 2.3. Any eight-term 4 × 3 simple biquadr atic form has sos( P ) ≤ 6 . Any nine-term 4 × 3 simple biquadr atic form has sos( P ) ≤ 7 . 3 Pr o of. F or an eight-term form, Lemma 2.2 guaran tees a 4-cycle C ij k l . The corresponding four terms can b e written as t wo squares via the identit y ab ov e, and the remaining four terms are individual squares, giving at most six squares. F or a nine-term form, the graph has nine edges and therefore still contains a 4-cycle. Applying the same decomp osition yields 2 + 5 = 7 squares. W e now recall the classical theorem of Hurwitz on sums of squares. Prop osition 2.4 (Hurwitz-t yp e decomp osition for 3 × 3 all-ones form) . The nine-term form Q 3 , 3 ( x , y ) = P 3 i =1 P 3 j =1 x 2 i y 2 j satisfies sos( Q 3 , 3 ) ≤ 4 . Pr o of. The follo wing explicit decomposition uses only four squares: Q 3 , 3 ( x , y ) = ( x 1 y 1 + x 2 y 2 + x 3 y 3 ) 2 + ( x 2 y 3 − x 3 y 2 ) 2 + ( x 3 y 1 − x 1 y 3 ) 2 + ( x 1 y 2 − x 2 y 1 ) 2 . Expanding verifies that all cross terms cancel, yielding precisely the nine square terms. This iden tity is a manifestation of Hurwitz’s theorem [6] on the comp osition of quadratic forms; it also corresponds to the norm in the quaternions. The structure of bipartite graphs with ten or elev en edges is more constrained. The follo wing structural fact is prov ed in [5, Lemma 4.3]: every 4 × 3 bipartite graph with ten edges either con tains tw o vertex-disjoin t C 4 cycles or con tains a complete bipartite subgraph K 3 , 3 (on three of the four ro ws and all three columns). With elev en edges the graph necessarily con tains a K 3 , 3 . Prop osition 2.5. A ny ten-term 4 × 3 simple biquadr atic form has sos( P ) ≤ 6 . Any eleven-term 4 × 3 simple biquadr atic form has sos( P ) ≤ 6 . Pr o of. Let P b e a simple form corresponding to a bipartite graph G with | E | edges. T en-term case ( | E | = 10 ). By the structural lemma from [5], tw o p ossibilities occur. • Two indep endent C 4 ’s. Let the t wo 4-cycles b e C 1 and C 2 , eac h using four distinct v ertices. Their edge sets are disjoint, and together they account for eight edges. The re- maining tw o edges are isolated (they cannot create another 4-cycle without using vertices already in the cycles). Each 4-cycle contributes tw o squares (b y the identit y for P ij k l ), and the t w o remaining edges are single squares. Hence sos( P ) ≤ 2 + 2 + 2 = 6. • Contains a K 3 , 3 . The K 3 , 3 subgraph (on three rows and all three columns) has nine edges. By Prop osition 2.4 this blo ck can b e written as four squares. The remaining single edge is a square term by itself. Thus sos( P ) ≤ 4 + 1 = 5 ≤ 6. Elev en-term case ( | E | = 11 ). Only one edge is missing from the complete bipartite graph K 4 , 3 . Let the missing edge b e ( p, q ). The three rows differen t from p together with all three 4 columns form a K 3 , 3 (all nine edges presen t). By Prop osition 2.4 this 3 × 3 blo c k contributes four squares. The remaining edges are those inciden t to v ertex p except the missing one; there are exactly t w o suc h edges (since p would b e adjacen t to the t wo columns other than q ). Eac h is a single square. Therefore sos( P ) ≤ 4 + 2 = 6. Com bining the results ab o ve we obtain the maxim um SOS rank for 4 × 3 simple forms. Theorem 2.6. The maximum SOS r ank of 4 × 3 simple biquadr atic forms is 7 . Pr o of. F rom Prop ositions 2.3 and 2.5, the SOS ranks for forms with 1-11 terms are b ounded as follo ws: • forms with up to 7 terms achiev e rank equal to the n umber of terms; • eight-term forms hav e rank at most 6; • nine-term forms ha ve rank at most 7; • ten- and elev en-term forms ha v e rank at most 6. The form P 4 , 3 , 7 in Theorem 2.1 attains rank 7, establishing the maximum. Hence, to find a 4 × 3 SOS biquadratic form with SOS rank greater than 7, we must consider non-simple forms. 3 y -Deficien t Biquadratic F orms In this section w e consider a natural sub class of biquadratic forms that admits a strong bound on the SOS rank via a splitting argumen t. Unlike simple forms, these forms may con tain cross terms, but they are constrained in a w ay that allows us to reduce the problem to smaller formats. Definition 3.1. A biquadr atic form P ( x , y ) in variables x = ( x 1 , . . . , x m ) and y = ( y 1 , . . . , y n ) is c al le d y -deficient if ther e exists an index j 0 ∈ { 1 , . . . , n } such that no term of P c ontains the variable y j 0 to gether with any other y j ( j  = j 0 ) or any pr o duct x i x k with i  = k . Equivalently, every term involving y j 0 must b e of the pur e squar e form x 2 i y 2 j 0 . In other w ords, P can b e written as P ( x , y ) = P 1 ( x , y ′ ) + y 2 j 0 T ( x ) , where y ′ denotes the vector of the remaining n − 1 v ariables, P 1 is a biquadratic form in x and y ′ (whic h may contain arbitrary cross terms among those v ariables), and T ( x ) = m X i =1 a i x 2 i , a i ≥ 0 , 5 is a sum of squares of linear forms in x (in fact a diagonal quadratic form). W e fo cus on the case m = 4, n = 3. Without loss of generality , let the deficient v ariable b e y 3 . Then P ( x , y ) = P 1 ( x , y 1 , y 2 ) + y 2 3 4 X i =1 a i x 2 i , a i ≥ 0 , where P 1 is a 4 × 2 biquadratic form. Theorem 3.2. Every 4 × 3 PSD y -deficient biquadr atic form (with deficiency in y 3 ) satisfies sos( P ) ≤ 9 . Pr o of. W rite P = P 1 + y 2 3 T as ab o ve. Setting y 3 = 0 sho ws that P 1 is PSD. A classical result of Calder n [3] states that ev ery m × 2 PSD biquadratic form is SOS; moreov er, by [1] w e ha v e BSR(4 , 2) = 5. Hence there exist bilinear forms ℓ 1 , . . . , ℓ 5 in x and ( y 1 , y 2 ) suc h that P 1 ( x , y 1 , y 2 ) = 5 X k =1 ℓ k ( x , y 1 , y 2 ) 2 . The term y 2 3 T is already a sum of squares: y 2 3 4 X i =1 a i x 2 i = 4 X i =1  √ a i x i y 3  2 . Therefore P = 5 X k =1 ℓ k ( x , y 1 , y 2 ) 2 + 4 X i =1  √ a i x i y 3  2 , whic h is a sum of at most 5 + 4 = 9 squares. Hence sos( P ) ≤ 9. Remark 3.3. The b ound 9 is the same as we would obtain for diagonal forms, which ar e a sp e cial c ase of y -deficient forms (any diagonal form satisfies the c ondition for every choic e of j 0 ). F or diagonal forms, an alternative pr o of c an b e given by splitting off the fourth r ow inste ad of a c olumn: write P = P 123 + R wher e P 123 is the 3 × 3 subform on r ows 1 - 3 and R involves only r ow 4 . Then by BSR(3 , 3) = 6 fr om [2], P 123 c an b e expr esse d with at most six squar es, and R c ontributes thr e e squar es, again yielding sos( P ) ≤ 9 . This alternative pr o of highlights the flexibility of the splitting te chnique. Remark 3.4. If the deficient variable wer e y 1 or y 2 , the same pr o of applies after r enaming the variables. Thus The or em 3.2 holds for any 4 × 3 PSD biquadr atic form that is deficient in some y -variable. Remark 3.5. The definition of y -deficiency is not symmetric in x and y . One might attempt to define an analo gous class of x -deficient forms, wher e a distinguishe d variable x i 0 app e ars only in pur e squar e terms x 2 i 0 y 2 j . Such forms would admit a de c omp osition P = P 1 ( x ′ , y ) + x 2 i 0 U ( y ) , 6 wher e P 1 is a 3 × 3 form. However, by the classic al r esult of Choi [4], not every 3 × 3 PSD biquadr atic form is SOS. Ther efor e, without additional assumptions, we c annot guar ante e an SOS de c omp osition for P 1 , and the appr o ach use d for y -deficient forms do es not extend. This asymmetry r efle cts the fundamental differ enc e b etwe en m × 2 forms (always SOS by Calder n ’s the or em [3]) and 3 × 3 forms (wher e c ounter examples exist). 3.1 An Example Illustrating the Bound Theorem 3.2 establishes an upp er bound of 9 for all 4 × 3 y -deficient biquadratic forms. T o illustrate that this b ound could b e sharp (i.e., that there ma y exist forms with SOS rank exactly 9), w e presen t a concrete diagonal example. Diagonal forms are a sp ecial case of y -deficient forms (they satisfy the deficiency condition for an y c hoice of the distinguished v ariable). Consider the 4 × 3 diagonal form P ( x , y ) = 3 X i =1 3 X j =1 a ij x 2 i y 2 j + 3 X j =1 b j x 2 4 y 2 j , with coefficients ( a ij ) 1 ≤ i,j ≤ 3 =    1 2 1 3 7 1 1 1 2    , ( b 1 , b 2 , b 3 ) = (1 , 1 , 1) . The 3 × 3 blo ck is exactly the matrix used in [7, Theorem 3.1] to illustrate a splitting argu- men t; there it w as shown that this block admits a decomp osition into seven squares. How ev er, b y the muc h stronger result BSR(3 , 3) = 6 from [2], every 3 × 3 SOS form can b e expressed with at most six squares. Consequently , w e can obtain a decomp osition of the whole 4 × 3 form in to at most nine squares using the alternativ e ro w?splitting pro of men tioned in Remark 3.2. Prop osition 3.6. The diagonal form define d ab ove satisfies sos( P ) ≤ 9 . Pr o of. Let Q ( x ′ , y ) = P 3 i =1 P 3 j =1 a ij x 2 i y 2 j b e the 3 × 3 diagonal subform, where x ′ = ( x 1 , x 2 , x 3 ). Since all a ij > 0, Q is p ositiv e semidefinite and hence a sum of squares. By BSR(3 , 3) = 6 [2], there exist bilinear forms ℓ 1 , . . . , ℓ 6 in x ′ and y suc h that Q ( x ′ , y ) = 6 X k =1 ℓ k ( x ′ , y ) 2 . The remaining terms inv olving x 4 are eac h a single square: 3 X j =1 b j x 2 4 y 2 j = 3 X j =1  p b j x 4 y j  2 . Hence P = 6 X k =1 ℓ k ( x ′ , y ) 2 + 3 X j =1  p b j x 4 y j  2 , whic h is a sum of nine squares. Th us sos( P ) ≤ 9. 7 Remark 3.7. The b ound sos( P ) ≤ 9 for y -deficient forms is not ne c essarily sharp within the sub class of diagonal forms, but the example in Pr op osition 3.6 do es not settle this question. A lthough the form is diagonal, it is not simple (the c o efficients ar e arbitr ary p ositive numb ers, wher e as simple forms as define d in Se ction 2 c orr esp ond to bip artite gr aphs with unit c o effi- cients). Ther efor e its SOS r ank c ould p otential ly exc e e d the maximum for simple forms, and it might even attain the b ound 9 if the 3 × 3 subform Q has SOS r ank 6 and the thr e e terms involv- ing x 4 c annot b e absorb e d into a mor e efficient de c omp osition. Whether such a form actual ly achieves r ank 9 r emains an op en pr oblem. The question of whether ther e exists a 4 × 3 y -deficient form (ne c essarily non-simple) with SOS r ank exactly 9 is inter esting for futur e r ese ar ch. Such a form would r e quir e that the 4 × 2 c omp onent P 1 has SOS r ank 5 (the maximum for 4 × 2 forms) and that the four squar es fr om the deficient variable c annot b e absorb e d into the de c omp osition of P 1 . 4 An Eigh t-Square F orm and a New Lo w er Bound for BSR(4 , 3) The simple form P 4 , 3 , 7 attains SOS rank 7 (Theorem 2.1), giving the lo w er bound BSR(4 , 3) ≥ 7. T o inv estigate whether higher ranks are p ossible, w e consider a natural mo dification of P 4 , 3 , 7 b y adding a carefully c hosen square term. The follo wing form will be sho wn to require exactly eigh t squares, establishing the impro ved b ound BSR(4 , 3) ≥ 8. Define Q ( x , y ) = P 4 , 3 , 7 ( x , y ) + ( x 4 y 2 + x 1 y 3 ) 2 , where P 4 , 3 , 7 = x 2 1 y 2 1 + x 2 2 y 2 2 + x 2 3 y 2 3 + x 2 1 y 2 2 + x 2 2 y 2 3 + x 2 3 y 2 1 + x 2 4 y 2 1 . Expanding ( x 4 y 2 + x 1 y 3 ) 2 = x 2 4 y 2 2 + x 2 1 y 2 3 + 2 x 1 x 4 y 2 y 3 , w e see that Q contains the nine pure square terms (1 , 1) , (1 , 2) , (1 , 3) , (2 , 2) , (2 , 3) , (3 , 1) , (3 , 3) , (4 , 1) , (4 , 2) eac h with co efficient 1, together with the single cross term 2 x 1 x 4 y 2 y 3 . Theorem 4.1 (An eigh t-square form and a new low er b ound for BSR(4 , 3)) . The form Q ( x , y ) = P 4 , 3 , 7 ( x , y ) + ( x 4 y 2 + x 1 y 3 ) 2 , wher e P 4 , 3 , 7 = x 2 1 y 2 1 + x 2 2 y 2 2 + x 2 3 y 2 3 + x 2 1 y 2 2 + x 2 2 y 2 3 + x 2 3 y 2 1 + x 2 4 y 2 1 , satisfies sos( Q ) = 8 . Conse quently, BSR(4 , 3) ≥ 8 . 8 Pr o of. Upp er b ound (explicit decomp osition). The follo wing eight squares repro duce Q exactly: Q = ( x 1 y 1 ) 2 + ( x 4 y 1 ) 2 + ( x 1 y 2 ) 2 + ( x 1 y 3 + x 4 y 2 ) 2 + ( x 2 y 2 ) 2 + ( x 2 y 3 ) 2 + ( x 3 y 1 ) 2 + ( x 3 y 3 ) 2 . A direct expansion confirms that all terms matc h: the first three squares giv e x 2 1 y 2 1 + x 2 4 y 2 1 + x 2 1 y 2 2 ; the term ( x 1 y 3 + x 4 y 2 ) 2 supplies x 2 1 y 2 3 + x 2 4 y 2 2 and the unique cross term 2 x 1 x 4 y 2 y 3 ; the remaining four squares provide x 2 2 y 2 2 + x 2 2 y 2 3 + x 2 3 y 2 1 + x 2 3 y 2 3 . No other pure squares or cross terms app ear. Hence sos( Q ) ≤ 8. Lo wer b ound (imp ossibilit y of sev en squares). Assume, for contradiction, that Q = P 7 k =1 ℓ k ( x , y ) 2 with bilinear forms ℓ k = 4 X i =1 3 X j =1 a ( k ) ij x i y j , a ( k ) ij ∈ R . Expanding and comparing co efficients with the explicit form of Q yields: • F or the nine presen t pure square terms, P 7 k =1 ( a ( k ) ij ) 2 = 1. • F or the three missing pure square terms (2 , 1) , (3 , 2) , (4 , 3), we must ha ve a ( k ) ij = 0 for all k . Th us, the corresp onding co efficien t vectors are zero: v 21 = 0 , v 32 = 0 , v 43 = 0 . • The only cross term is 2 x 1 x 4 y 2 y 3 , so P 7 k =1 a ( k ) 13 a ( k ) 42 = 1, while all other cross co efficien ts v anish. Define v ectors v ij = ( a (1) ij , . . . , a (7) ij ) ∈ R 7 for the nine presen t index pairs. Then ∥ v ij ∥ = 1 and v 13 · v 42 = 1. By the Cauch y-Sch w arz inequalit y , v 13 = v 42 ; w e denote this common unit v ector by w : w := v 13 = v 42 . W e now sho w that the set of eigh t v ectors S = { w , v 11 , v 12 , v 22 , v 23 , v 31 , v 33 , v 41 } is m utually orthogonal in R 7 . The orthogonality follo ws from the v anishing co efficien ts of absent monomials x i x p y j y q , whic h imply v ij · v pq + v iq · v pj = 0. Cate gory A: Dir e ct Ortho gonality (14 p airs). These arise from absen t mixed square terms where v ectors share a ro w or column: • w ⊥ { v 11 , v 12 } via absent x 2 1 y 1 y 3 , x 2 1 y 2 y 3 ; w ⊥ { v 22 , v 33 } via absent x 2 x 4 y 2 2 , x 1 x 3 y 2 3 (using w = v 42 , v 13 ); w ⊥ v 41 via absen t x 2 4 y 1 y 2 . • v 11 ⊥ { v 12 , v 31 , v 41 } via absen t x 2 1 y 1 y 2 , x 1 x 3 y 2 1 , x 1 x 4 y 2 1 . • v 12 ⊥ v 22 via absen t x 1 x 2 y 2 2 ; v 22 ⊥ v 23 via absen t x 2 2 y 2 y 3 . 9 • v 23 ⊥ v 33 via absen t x 2 x 3 y 2 3 ; v 33 ⊥ v 31 via absen t x 2 3 y 1 y 3 . • v 31 ⊥ v 41 via absen t x 3 x 4 y 2 1 . Cate gory B: Indir e ct Ortho gonality via Zer o V e ctors (14 p airs). These arise from absent cross terms where the complementary pair inv olv es a zero v ector ( v 21 , v 32 , or v 43 ): • w ⊥ v 31 : F rom x 3 x 4 y 1 y 2 = 0 = ⇒ v 31 · w + v 32 · v 41 = 0 (since v 32 = 0 ). • v 11 ⊥ { v 22 , v 23 } : F rom x 1 x 2 y 1 y 2 = 0 , x 1 x 2 y 1 y 3 = 0 using v 21 = 0 . • v 11 ⊥ v 33 : F rom x 1 x 3 y 1 y 3 = 0 = ⇒ v 11 · v 33 + w · v 31 = 0; since w ⊥ v 31 (ab o ve), v 11 ⊥ v 33 . • v 12 ⊥ { v 31 , v 33 } : F rom x 1 x 3 y 1 y 2 = 0 , x 1 x 3 y 2 y 3 = 0 using v 32 = 0 . • v 12 ⊥ v 41 : F rom x 1 x 4 y 1 y 2 = 0 = ⇒ v 11 · w + v 12 · v 41 = 0; since v 11 ⊥ w , v 12 ⊥ v 41 . • v 12 ⊥ v 23 : F rom x 1 x 2 y 2 y 3 = 0 = ⇒ v 12 · v 23 + w · v 22 = 0; since w ⊥ v 22 (Cat A), v 12 ⊥ v 23 . • v 22 ⊥ { v 31 , v 33 , v 41 } : F rom x 2 x 3 y 1 y 2 = 0 , x 2 x 3 y 2 y 3 = 0 , x 2 x 4 y 1 y 2 = 0 using v 21 = 0 or v 32 = 0 . • v 23 ⊥ { v 31 , v 41 } : F rom x 2 x 3 y 1 y 3 = 0 , x 2 x 4 y 1 y 3 = 0 using v 21 = 0 or v 43 = 0 . • v 33 ⊥ v 41 : F rom x 3 x 4 y 1 y 3 = 0 using v 43 = 0 . Th us, all 8 v ectors in S are nonzero and pairwise orthogonal in R 7 . This implies they are linearly indep endent, whic h is imp ossible in a 7-dimensional space. The contradiction pro ves that no 7-square decomp osition exists. Therefore, sos( Q ) = 8. Remark 4.2. The form Q is not simple (it c ontains the cr oss term 2 x 1 x 4 y 2 y 3 ) and is not y - deficient. It demonstr ates that p erturbing the extr emal simple form P 4 , 3 , 7 c an incr e ase the SOS r ank, r aising the lower b ound for BSR(4 , 3) fr om 7 to 8 . Remark 4.3. The c onstruction shows that the extr emal simple form P 4 , 3 , 7 c an b e p erturb e d by a c ar eful ly chosen squar e to incr e ase the SOS r ank by one, while a differ ent p erturb ation (such as ( x 1 y 3 + x 2 y 1 ) 2 ) left the r ank unchange d. This sensitivity suggests that the exact value of BSR(4 , 3) may lie strictly b etwe en 8 and 11 , and further investigation of non-simple, non- y - deficient forms is ne e de d. 10 5 Conclusion In this pap er, we hav e inv estigated the maxim um sum-of-squares rank for t wo imp ortan t sub- classes of 4 × 3 biquadratic forms: simple forms and y -deficient forms. Our main results pro vide exact determinations and impro ved b ounds that significantly narrow the gap b etw een the kno wn lo wer b ound z (4 , 3) = 7 and the general upp er bound mn − 1 = 11 for arbitrary 4 × 3 SOS biquadratic forms. F or simple biquadratic forms, we completely c haracterized the extremal case. By analyzing the underlying bipartite graph structure and exploiting the relationship b et ween C 4 -free graphs and SOS rank established in [5], we pro v ed in Theorem 2.6 that the maxim um ac hiev able SOS rank is exactly 7, attained by the form P 4 , 3 , 7 . This settles the question for simple forms and demonstrates that the lo wer bound from the Zarankiewicz num b er is sharp within this subclass. F or y -deficient biquadratic forms, a class that allows cross terms among t w o of the three y -v ariables, we prov ed in Theorem 3.2 an upp er b ound of 9. The pro of combines Calder n’s theorem on m × 2 forms [3] with the kno wn v alue BSR(4 , 2) = 5 from [1]. These results sho w that the presence of cross terms do es not necessarily increase the SOS rank, provided they are confined to a 4 × 2 subsystem. Our primary contribution is Theorem 4.1, where w e constructed an explicit 4 × 3 biquadratic form Q and prov ed that sos( Q ) = 8. This improv es the general low er b ound for BSR(4 , 3) from 7 to 8 and represen ts the first constructive progress on this problem in recent years. The form Q is neither simple nor y -deficient, confirming that any form achieving a rank greater than 8 m ust lie outside these t wo sub classes. The results highlight an imp ortan t phenomenon: the maxim um SOS rank for 4 × 3 bi- quadratic forms dep ends crucially on the algebraic structure. Simple forms cannot exceed rank 7; y -deficient forms satisfy rank at most 9; and we hav e now sho wn that rank 8 is ac hiev able. The in terv al of possible v alues for BSR(4 , 3) is no w 8 ≤ BSR(4 , 3) ≤ 11 . Sev eral directions for future researc h emerge naturally from this work: 1. Closing the gap for y -deficien t forms. The question of whether there exists a 4 × 3 y -deficient form with SOS rank exactly 9 remains op en. F or diagonal forms this reduces to the existence of a 3 × 3 diagonal form with SOS rank exactly 6. If suc h a form exists, then our b ound is sharp; if all 3 × 3 diagonal forms hav e rank at most 5, then the maximum for 4 × 3 diagonal forms would be at most 8. 2. Exploring other structured sub classes. Bey ond simple and y -deficient forms, one could in v estigate forms with other sparsity patterns, such as those corresp onding to bi- partite graphs with prescrib ed girth or degree constrain ts. The in terplay b etw een graph- theoretic properties and SOS rank remains a rich area for exploration. 11 3. T ow ards the determination of BSR(4 , 3) . The ultimate goal is to determine the maxim um SOS rank for arbitrary 4 × 3 biquadratic forms. Our results show that an y form ac hieving rank greater than 7 m ust b e non-simple, and any form ac hieving rank greater than 9 m ust b e neither diagonal nor y -deficient. With the low er b ound now raised to 8, the cen tral op en problem is to determine whether the true v alue is 8, 9, 10, or 11. Constructing explicit examples with ranks 9, 10, or 11 or proving that suc h ranks are impossible remains a c hallenging but essen tial task. 4. Extension to larger formats. The methods developed here, particularly the use of splitting arguments and kno wn b ounds for smaller formats (3 × 3 , 4 × 2), ma y generalize to m × n forms with larger m, n . The Zarankiewicz num b er z ( m, n ) pro vides a low er b ound, but the exact maxim um SOS rank for larger formats remains largely unexplored. In summary , this paper pro vides a comprehensiv e analysis of SOS ranks for t wo fundamen tal sub classes of 4 × 3 biquadratic forms, establishing exact results for simple forms and improv ed b ounds for y -deficien t forms. Most imp ortantly , it in tro duces the first constructive improv ement to the general lo wer b ound in decades, proving that BSR(4 , 3) ≥ 8. These findings con tribute to the broader program of understanding the sum-of-squares represen tation of biquadratic forms and their connections to com binatorial structures suc h as bipartite graphs and Zarankiewicz n umbers. Ac knowledgemen t W e are thankful to Professor Greg Blekherman who told us that he b e- liev es B S R (4 , 3) ≥ 8. This work w as partially supp orted by Research Center for In telligent Op erations Research, The Hong Kong Polytec hnic Universit y (4-ZZT8), the National Natural Science F oundation of China (Nos. 12471282 and 12131004), and Jiangsu Pro vincial Scientific Researc h Center of Applied Mathematics (Gran t No. BK20233002). Data av ailability No datasets w ere generated or analysed during the current study . Conflict of in terest The authors declare no conflict of in terest. References [1] G. Blekherman, D. Plaumann, R. Sinn and C. Vinzan t, “Lo w-rank sum-of-squares rep- resen tations on v arieties of minimal degree”, In ternational Mathematics Researc h Notices 2019 (2019) 33-54. [2] G. Blekherman, R. Sinn, G. Smith and M. V elasco, “Sums of squares and quadratic p er- sistence on real pro jective v arieties”, Journal of the Europ ean Mathematical So ciet y 24 (2021) 925-965. 12 [3] A.P . Calder´ on, “A note on biquadratic forms”, Linear Algebra and Its Applications 7 (1973) 175-177. [4] M.-D. Choi, “Positiv e semidefinite biquadratic forms”, Linear Algebra and Its Applications 12 (1975) 95-100. [5] C. Cui, L. Qi and Y. Xu, “The Sum of squares rank of biquadratic forms and the Zarankiewicz n umber”, F eburary 2026, [6] A. Hurwitz, “ ¨ Ub er die Comp osition der quadratisc hen F ormen”, Mathematische Annalen 50 (1898), 177-188. [7] L. Qi, C. Cui and Y. Xu, “Sum of squares decomp ositions and rank b ounds for biquadratic forms”, Mathematics 14 (2026) No. 635. 13