Secant Degeneracy Index of the Standard Strata in The Space of Binary Forms

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1 Arnold Math J. RESEARCH CONTRIBUTION Secant Deeneracy Index of the Standard Strata in The Space of Binary Forms Gleb Nenashev 1 Boris Shapiro 1 Michael Shapiro 2 Received: 30 January 2017 / Revised: 3 November 2017 / Accepted: 7 November 2017 The Author(s) This article is an open access publication Abstract The space Pol d CP d of all complex-valued binary forms of deree d (considered up to a constant factor) has a standard stratification, each stratum of which contains all forms whose set of multiplicities of their distinct roots is iven by a fixed partition μ d. For each such stratum S μ, we introduce its secant deeneracy index l μ which is the minimal number of projectively dependent pairwise distinct points on S μ, i.e., points whose projective span has dimension smaller than l μ 1. In what follows, we discuss the secant deeneracy index l μ and the secant deeneracy index l μ of the closure S μ. Keywords Strata of the classical discriminant Secant varieties Partitions Mathematics Subject Classification Primary 14M99; Secondary 11E25 11P05 1 Introduction Below by a form we will always mean a binary form. The standard stratification of the d-dimensional projective space Pol d of all complex-valued binary forms of deree d (considered up to a non-vanishin constant factor) accordin to the multiplicities of B Boris Shapiro shapiro@math.su.se Gleb Nenashev nenashev@math.su.se Michael Shapiro mshapiro@math.msu.edu 1 Department of Mathematics, Stockholm University, Stockholm, Sweden 2 Department of Mathematics, Michian State University, East Lansin, MI , USA

2 G. Nenashev et al. their distinct roots is a well-known and widely used construction in mathematics (see e.. Arnol d 2014; Vassiliev 1992; Khesin and Shapiro 1992). Its strata denoted by S μ are enumerated by partitions μ d. In particular, cohomoloy of S μ with different coefficients appears in many topoloical problems and was intensively studied over the years, see e.. Vassiliev (1992) and references therein. Definition 1 Given a positive-dimensional quasi-projective variety V CP d,we define its secant deeneracy index l V as the minimal positive inteer l such that there exist l distinct points on V which are projectively dependent, i.e. whose projective span has dimension at most l 2. (Observe that sinular points of V are not considered as collapsin distinct points. For example, accordin to our definition, a point of selfintersection of V is still considered a just one point of V ). Remark 1 The secant deeneracy index in much more eneral context has been introduced by e.. Beltrametti and Sommese (1995) while discussin the l-ampleness of the linear system of hyperplane sections for different l. This notion was further developed in a recent paper (Chiantini and Ciliberto 2010). (We have to mention that unlike many modern authors, we only consider reduced finite subschemes of V ). As was pointed out to us by the anonymous referee, the secant deeneracy index has already appeared in a number of topics in alebraic eometry and, in particular, is closely connected with the identifiability of tensors and hiher order normality. For example, a related question about the uniqueness of representation of eneric forms of subeneric rank as sums of powers of linear forms has been studied in a recent article (Chiantini et al. 2015). Remark 2 Obviously, 3 l V d + 2. The upper bound is attained for a rational normal curve in CP d. On the other hand, if V contains CP 1 \{finite set}, then l V = 3. We owe to the anonymous referee the important observation that the above trivial upper bound l V = d + 2 is attained only on (Zariski open subsets of) rational normal curves and this bound can be improved as follows. Namely, if c := codim V = d dim V, then l V c + 2 unless V is a variety of minimal deree which in our notation means that de V = c + 1. By the classical results of Del Pezzo and Bertini, every positivedimensional variety of minimal deree contains a line which implies that l V = 3, unless V is a rational normal curve or a Veronese surface, see e.. Theorem 1 of Eisenbud and Harris (1987). For the Veronese surface (which is a surface in CP 5 ), l V = 4 = c + 1. To see that, one can take 4 points on a conic. Thus, one concludes that l V c + 2 unless V is a Zariski open subset of a rational normal curve in which case l V = c + 3 = d + 2. Observe that if a quasi-projective variety V is contained in quasi-projective W, then l V l W. For a positive-dimensional quasi-projective variety V CP d, denote by l V the secant deeneracy index of the closure V CP d. Obviously, l V l V.The latter inequality can be strict as shown by Example 1 below. The principal question considered in the present paper is as follows. Problem 1 For a iven partition μ d, calculate/estimate its secant deeneracy indices l μ := l Sμ and l μ := l S μ.

3 Secant Deeneracy Index of The Standard Strata... Example 1 For μ(d) = (2d + 1, d, d, d, d) and μ (d) = (2d + 1, 2d, 2d), l μ(d) = l μ (d) = 3, but l μ(d) rows to infinity when d. (This result follows from Theorem 1 below.) For a iven partition μ, the equation f 1 + f f lμ = 0, (1) is called the minimal secant deeneracy relation for S μ. A solution of the latter equation is a collection of pairwise non-proportional forms from S μ satisfyin (1). Analoously, for a iven partition μ, the equation f 1 + f f l μ = 0, (2) is called the minimal secant deeneracy relation for S μ. A solution of the latter equation is a collection of pairwise non-proportional forms from S μ satisfyin (2). Most of our results deal with the secant deeneracy index l μ. However, the second part of Theorem 1 provides a non-trivial lower bound for l μ eneralizin a similar result of a well-known paper Newman and Slater (1979) from 1979 where the special case of partitions with equal parts was considered. The first result of this note is as follows. Recall the notion of the refinement partial order on the set of all partitions of a iven positive inteer d. Namely, μ μ in this order if μ is obtained from μ by merin of some parts of μ. The unique minimal element of this partial order is (1) d, while its unique maximal element is (d). For a partition μ = (μ 1 μ 2 μ r ), define its jump multiset J μ as the multiset of all positive numbers in the set {μ 1 μ 2,...,μ r 1 μ r,μ r }. We denote by h μ the minimal (positive) jump of μ, i.e. the minimal element of J μ, and by h μ the minimal jump of all partitions μ μ. Theorem 1 For any μ = (μ 1 μ 2 μ r ), (i) l μ > h μ , and (ii) l μ > h μ To formulate further results, we divide the set of all partitions into two natural disjoint subclasses as follows. Notation. For a iven partition μ = (μ 1 μ 2 μ r ) and a non-neative inteer t, define the partition μ t as μ t := (μ 1 + t μ 2 + t μ r + t). Definition 2 We say that a partition μ has a rowin secant deeneracy index if lim t l μ t =+ and we say that μ has a stabilisin secant deeneracy index otherwise.

4 G. Nenashev et al. We are able to characterize these two classes in the followin terms. Definition 3 Given a partition μ and a positive inteer m, a solution of f 1 + f f m = 0, (3) with pairwise non-proportional f i S μ is called a common radical solution if all f i s have the same radical, i.e. the same set of distinct linear factors (considered up to a constant factor). We call a partition μ such that there exists m and a common radical solution of (3)apartition admittin a common radical solution. The followin proposition is straihtforward. Proposition 2 A partition μ = (μ 1 μ 2 μ r ) = (i m 1 1, i m 2 2,...,i m s s ) with distinct i j s has a stabilisin secant deeneracy index if and only if, for some positive inteer m, there exists a common radical solution of (3). A partition μ as above has a rowin secant deeneracy index if and only if the linear span of the Sym r -orbit of r! any form f S μ has the dimension equal to the multinomial coefficient m 1!m 2!...m s!. (Here the symmetric roup Sym r acts on any f S μ by permutin all its r distinct roots.) At the moment we do not have a purely combinatorial description of partitions admittin a common radical solution. However we were able to study a somewhat stroner property. Definition 4 We say that a partition μ = (μ 1 μ 2 μ r ) admits a stronly common radical solution if for some inteer m, a common radical solution exists for any choice of r distinct roots. Theorem 3 A partition μ = (μ 1 μ 2 μ r ) admits a stronly common radical solution if there exists a sequence {a 1,...,a r } of positive inteers such the number of different permutations π of μ such that (π μ) i a i is at least μ ri=1 a i + 2, where (π μ) i is the i-th entry of the partition (π μ). In fact, we stronly suspect that the converse to Theorem 3 holds as well. Conjecture 1 A necessary and sufficient condition for a partition μ = (μ 1 μ 2 μ r ) to admit a stronly common radical solution is iven by the existence of a sequence {a 1,...,a r } of positive inteers such the number of different permutations π of μ such that (π μ) i a i is at least μ r i=1 a i + 2, where (π μ) i is the i-th entry of the partition (π μ). At the moment we can settle Conjecture 1 for a lare class of partitions, but not for all partitions. The structure of the paper is as follows. In Sect. 2, we formulate several eneral results about l μ, the most interestin of them bein an upper bound of l μ in terms of the minimal jump. In Sect. 3, we discuss common radical solutions of (1) and in Sect. 4, we present a number of open problems.

5 Secant Deeneracy Index of The Standard Strata... 2 General Results on the Secant Deeneracy Index Given a partition μ = (μ 1 μ 2 μ r ), we call ν = (μ i1 μ i2 μ is ) where 1 i 1 < i 2 < < i s r,asubpartition of μ. Proposition 4 For a partition μ = (μ 1 μ 2 μ r ) and any subpartition ν of μ, the inequality holds. In particular, l μ μ r + 2. l μ l ν, Proof Given a subpartition ν = (μ i1 μ i2 μ is ) of a partition μ = (μ 1 μ 2 μ r ),let f 1 + f f lν = 0, be a linear dependence of pairwise non-proportional binary forms from S ν realizin its secant deeneracy index. Take the partition μ = μ\ν = ( μ 1 μ 2 μ r s ). Multiplyin the latter equality by r s j=1 (x a j y) μ j, where a j are eneric complex numbers, we et a linear dependence between polynomials in S μ. The inequality l μ μ r + 2 is a special case of the eneral inequality, if one chooses ν = (μ r ). Observe that for the partition (d) d, l (d) = d + 2, since the set of binary forms of deree d with a root of multiplicity d is a rational normal curve in Pol d CP d. Example 2 The latter upper bound l μ μ r + 2 is sharp in case of any partition with μ r = 1, but not in eneral. Namely, already for μ = (2 2 ) 4, l μ = 3 < 4 = μ For μ = (3 2 ) 6, l μ = 4 < 5; μ = (4 2 ) 8, l μ = 4 < 6, see Reznick (1999). Before formulatin eneral results about l μ, let us present several concrete classes of μ and some information about the correspondin l μ. Proposition 5 Let μ = (μ 1 μ 2 μ r ) be a partition with two different indices i 1 and i 2 such that μ i1 μ i1 +1 = μ i2 μ i2 +1 = 1. Then, l μ 4. Proof Without loss of enerality, assume that i 1 < i 2, and consider two different cases. Case 1. i 2 = i Take a subpartition ν = (μ i1,μ i1 +1,μ i1 +2) = (μ i1 +2+2,μ i ,μ i1 +2) and set k = μ i1 +2. We know that l μ l ν. So it is enouh to prove that l ν 4. Take three distinct complex numbers p, q andr, and consider four polynomials 1 = (x p) k+2 (x q) k+1 (x r) k, 2 = (x p) k+2 (x r) k+1 (x q) k, 3 = (x q) k+2 (x p) k+1 (x r) k, 4 = (x r) k+2 (x p) k+1 (x q) k. A linear combination a 1 + b 2 + c 3 + d 4 is iven by Q(x)(a(x p)(x q) + b(x p)(x r) + c(x q) 2 + d(x r) 2 ),

6 G. Nenashev et al. where Q(x) = (x p) k+1 (x q) k (x r) k. Polynomials (x p)(x q), (x p)(x r), (x q) 2 and (x r) 2 are linearly dependent. Thus there exist a, b, c, d such that a 1 + b 2 + c 3 + d 4 = 0. Hence l ν 4. Case 2. i 2 > i Take a subpartition ν = (μ i1,μ i1 +1,μ i2,μ i2 +1) = (μ i ,μ i1 +1,μ i ,μ i2 +1); setk 1 = μ i1 +1 and k 2 = μ i2 +1. We know that l μ l ν. So it is enouh to prove that l ν 4. Take four distinct complex numbers p, q, r and t, and consider four polynomials 1 = (x p) k1+1 (x q) k 1 (x r) k2+1 (x s) k 2, 2 = (x q) k1+1 (x p) k 1 (x r) k2+1 (x s) k 2, 3 = (x p) k1+1 (x q) k 1 (x s) k2+1 (x r) k 2, 4 = (x q) k1+1 (x p) k 1 (x s) k2+1 (x r) k 2. A linear combination a 1 + b 2 + c 3 + d 4 is iven by R(x)(a(x p)(x r) + b(x q)(x r) + c(x p)(x s) + d(x q)(x s)), where R(x) = (x p) k 1(x q) k 1(x r) k 2(x s) k 2. Polynomials (x p)(x r), (x q)(x r), (x p)(x s) and (x q)(x s) are linearly dependent. Thus there exist a, b, c, d such that a 1 + b 2 + c 3 + d 4 = 0, and hence l ν 4. Definition 5 By the radical of a iven binary form we mean the binary form obtained as the product of all distinct linear factors of the oriinal form. Proposition 6 For any partition μ = (μ 1 μ 2 μ r ) and iven an arbitrary positive inteer i, consider the partition μ = (μ 1 + i μ 2 + i μ r + i i, i,...,i), where the entry i is repeated r(l μ 1) times at the end of μ. Then, l μ l μ. Proof Let f 1,..., f lμ be a solution of (1). Consider the radical of the polynomial f 1 f 2... f lμ. Since every form f j has exactly r distinct roots, the deree of is less than rl μ. Construct as the product of by rl μ de() new distinct linear forms, and set f j = f j ( ) i,for j = 1,...,l μ. It is easy to see that each f j has the root partition iven by μ. Furthermore, one has f f l μ = ( f f lμ ) ( ) i = 0, hence, l μ l μ. Corollary 1 For any partition μ containin the subpartition ν = (t + 1, t, t), where t is a positive inteer, the secant deeneracy index l μ equals 3. More enerally, for any positive inteer t, and any partition μ containin the subpartition ν = (t + i, t, t,...,t), the secant deeneracy index l }{{} μ is at most i + 2. i+1

7 Secant Deeneracy Index of The Standard Strata... We finish this section with the proof of Theorem 1. Proof Both parts of Theorem 1 are settled in a similar way. Namely, iven μ, let { f 1,..., f l } be a collection of forms solvin either (1)or(2). (In the first case l = l μ and in the second case l = l μ.) Assume that { f 1,..., f l } ives a counterexample to the statement. Denote by the GCD of { f 1,..., f l } and consider the relation f f l = 0. In case (i) of Theorem 1, for any i, every root of the polynomial f i has multiplicity at least h μ, because this multiplicity equals μ k μ l for some k l. Observe that, for all k l, μ k μ l is either 0 or is reater than or equal to h μ. In case (ii) of Theorem 1, for any i, every root of the polynomial f i has multiplicity at least h μ, because this multiplicity equals k A μ k l B μ l, where A and B are two subsets of {1,...,r}. Observe that, for all pairs (A, B ) such that A B =, k A μ k l B μ l is either 0 or is reater than or equal to h μ. The rest of the proof is the same in both cases. In what follows, h stands for h μ in case (i) and for h μ in case (ii). Consider the sequence of Wronskians w i = W ( f1,..., f i 1, f i+1,..., f ) l, i = 1,...,l. All these Wronskians are proportional to each other due to the latter relation. Let α be a root of some f i. There exists an index s such that f s is not divisible by (x α), since otherwise is not the GCD. For a iven t, consider the multiplicity of the root of w t at α. It satisfies the inequality: ord α (w t ) ( ( )) { f j ord α (l 2)# i : (x α) f } i, because any column of the Wronski matrix correspondin to (x α) f j by (x α) ord α Hence, ( f j ) l+2. de w 1 On the other hand, ( de w 1 (l 1) de l i=1 ( de ( fi = l( μ de ) (l 2) ( fi ) ( )) fi (l 2)# roots l i=1 ( ) fi # roots. is divisible ) ) l + 2 = (l 1)( μ de ) (l 1)(l 2).

8 G. Nenashev et al. We obtain (l 1)( μ de ) (l 1)(l 2) l( μ de ) (l 2) i.e., (l 2) l i=1 l i=1 ( ) fi # roots (l 1)(l 2) μ de. ( ) fi # roots, ( ) The number # fi roots of distinct roots is at most μ de h, because each root has multiplicity at least h. Thus (l 2)(l 1) μ de (l 1)(l 2) μ de. h μ Hence, (l 2)(l 1) >h. 3 Partitions with Growin and Stabilisin Secant Deeneracy Index Theorem 7 If μ = (μ 1 μ 2 μ r ) satisfies the inequality m μr r 1 + 1, then any solution of (3) is a common radical solution. Proof Assume the opposite. Let { f 1,..., f m } be a solution of (1) which is not a common radical solution. Let be the GCD of { f 1,..., f m }. For the term f i = c i (x a i,1 ) μ 1 (x a i,r ) μ r, define i := (x a i,1 ) μ 1 m+2 (x a i,r ) μ r m+2. Observe that i is a polynomial, because any root of f i has multiplicity at least μ r > m. Consider the sequence of Wronskians w i = W ( f 1,..., f i 1, f i+1,, f m ), i = 1,...,m. They are proportional to each other, because f f m = 0. Notice that, for i = t, the column in the Wronski matrix for w t correspondin to f i is divisible by i. Hence w t is divisible by m i=1 i / t. Since { f 1,..., f m } is not a common radical solution, there exists α C, such that α is a root of f p but is not a root of f q for some p = q. Since the Wronskians w p and w q are proportional, they are divisible by LCM ( mi=1 i p, mi=1 ) i = q mi=1 mi=1 i GCD( p, q ) = i p p GCD( p, q ).

9 Secant Deeneracy Index of The Standard Strata... Then these Wronskians are divisible by are reater than or equal to mi=1 i p (m 1)( μ r(m 2)) + μ r m + 2. (x α) μ r m+2. Therefore their derees On the other hand, the derees of the Wronskians are at most (m 1)( μ m + 2). Thus, (m 1)( μ m + 2) (m 1)( μ r(m 2)) + μ r m + 2, which implies that (m 1)(m 2) r(m 1)(m 2) + μ r m + 2. After straihtforward simplifications the latter inequality ives m 1 μr r 1. Contradiction. Corollary 2 For μ = (μ 1 μ 2 μ r ), either l μ μr r or any solution of (1) is a common radical solution. Remark 3 For any partition μ with a rowin secant deeneracy index, i.e., for l μ t, we know that μr + t r l μ t μ r + t + 2, see Proposition 4 and Theorem 1. Now we present a sufficient condition for μ to have a rowin secant deeneracy index. Corollary 3 Any partition μ = (μ 1 μ 2 μ r ), such that every its jump is at least (r!) 2, has a rowin secant deeneracy index. Proof Assume that l μ t does not row to infinity. Then by Theorem 1, l μ t h tμ h 0μ (r!) > r!. However the number of different polynomials (up to a constant factor) with fixed r roots of multiplicities μ t is at most r!. Hence no common radical solution can exist. Contradiction. We continue with the proof of Theorem 3.

10 G. Nenashev et al. Proof Let f = (x c 1 y) μ 1 (x c 2 y) μ 2... (x c r y) μ r be any form from S μ. Consider the set D (a1,...,a r ) of permutations of the multiset τ μ satisfyin the asumptions (i) and (ii) of Theorem 3. For any π D (a1,...,a r ), define f π as the form from the Sym r -orbit of f correspondin to π. Any such form f π is divisible by = (x c 1 y) a 1 (x c 2 y) a 2... (x c r y) a r, because π satisfies the above assumptions. For any π D (a1,...,a r ), define fˆ π := f π S μ ri=1 a i.if D (a1,...,a r ) μ ri=1 a i +2, then the forms fˆ π are linearly dependent. Therefore, the forms f π, where π runs over D (a1,...,a r ), are also linearly dependent. The next proposition shows that if there are many jumps of small sizes, then the secant deeneracy index is bounded. Proposition 8 Let d be a positive inteer reater than 45. For a partition μ = (μ 1 μ 2 μ r ), if the number of jumps of sizes less than or equal to d is at least 2(lo d + lo lo d + 2), then l μ d(lo d + lo lo d) + 2. (Herebylo we mean the binary loarithm, i.e. the loarithm with base 2.) Proof Assume that there are at least 2(lo d + lo lo d) + 2) such jumps. Consider every second such jump; the number of these jumps is at least t =[lo d+lo lo d+2]. Assume that they occupy positions j 1 <...< j t, i.e. (μ ji μ ji +1) d,fori [1, t]. Furthermore j i + 1 < j i+1, because there is a nontrivial jump between them. Consider the set of permutations of μ ={μ 1,...,μ r } such that: i / {j 1,..., j t }, π i μ i ; i {j 1,..., j t }, π i μ i+1. The number of such permutations is 2 t. By Theorem 3, there is a solution of the size t i=1 (μ ji μ ji +1) + 2 d t + 2, because d t + 2 d (lo d + lo lo d + 2) + 2 d (lo d + lo lo d + 3) d 2 lo d 2 lo d+lo lo d+1 2 t which finishes the proof. 3.1 Examples It is rather obvious that all partitions with two parts have rowin secant deeneracy index. Indeed, if there exists a common radical solution of (3), then its lenth m is smaller than or equal to r! which in case of two parts equals 2. Proposition 9 (i) For partitions μ = (p, 2), one has that l μ = 3 when p = 2, and l μ = 4 when p > 2. (ii) For partitions μ = (p, 3), one has that l μ = 4 when p = 3, 4, 5 and l μ = 5 when p > 7. Cases μ = (6, 3) and μ = (7, 3) are still open. Proof In case μ = (3, 4) we found an example: y 3 (x + y) 4 y 3 x 4 = L(x + ay) 3 y 4 + (1 L)(x + by) 3 y 4,

11 Secant Deeneracy Index of The Standard Strata... where a = 3 3, b = and L = ; In case μ = (5, 3) we found an example: f 1 + f 2 f 3 f 4 = 0, where f 1 (x) = (x + c1 5y)3 (x + c1 3 y)5 ; f 2 (x) = (x + c2 5y)3 (x + c2 3 y)5 ; f 3 (x) = (x + c1 5 y)3 (x + c1 3y)5 ; f 4 (x) = (x + c2 5 y)3 (x + c2 3y)5.Here c 1 = c 2 = ( 1 + i ) Usin computer alebra packaes we were able to prove the followin statement. Proposition 10 For a partition μ with three parts a b c, the followin three conditions are equivalent: (i) μ has a stabilisin secant deeneracy index; (ii) μ has a stronly stabilisin secant deeneracy index; (iii) the triple a b c belons to one of the followin three types: a = b, c = a+1; b = a + 1, c= a + 2;b= a + 1, c= a + 3. Proof (i) (ii). Note that any triple of distinct points in CP 1 can be transformed into any other triple of distinct points in CP 1 by a Möbius transformation. Therefore, if a stabilisin solution exists for some choice of three distinct roots, it exists for any other triple of distinct roots. (ii) (iii). Consider the S 3 -orbit of the function f 1 (t) = (t x 1 ) a (t x 2 ) b (t x 3 ) c containin 6 pairwise distinct functions f j, j = 1,...,6 in case a < b < c and 3 such functions if any two of them coincide. Denote by W the correspondin Wronskian of f 1,..., f 6 (divided by the trivial factor which is a polynomial of deree 6 in t). If μ = (a, b, c) has stronly stabilisin secant deeneracy index, then W 0for all t, x 1, x 2, x 3. In particular, all coefficients of W (t) should vanish identically. Usin Maple we able to find all possible triples (a, b, c) for which all coefficients of W (t) are identically 0. The solutions are presented in (iii) above. 4 Final Remarks and Problems 1. Conjecture 1 from the introduction is equivalent to the claim that for an appropriate choice of an r-tuple of distinct complex numbers, a certain matrix whose entries are polynomials in these numbers has full rank. In other words, that some maximal minor of this matrix is a non-trivial polynomial in the latter r-tuple which is hihly plausible. Unfortunately, the structure of the matrix is rather involved and so far we are only able to handle a lare number of special cases.

12 G. Nenashev et al. 2. Conditions formulated in Proposition 2 are difficult to check which motivates the followin questions. Problem 2 Give necessary and sufficient combinatorial conditions for a partition μ to have a rowin/stabilizin secant deeneracy index. Problem 3 For any partition μ with a rowin secant deeneracy index, what is the leadin term of the asymptotic of l μ t, when t +? Does it depend on a particular choice of μ? 3. The next statement is obvious. Proposition 11 The number l μ is monotone non-decreasin in the refinement partial order. In other words, if μ μ, then l μ l μ. Based on a substantial number of calculations, we conjecture the followin. Conjecture 2 For any partition μ d, there exists μ μ such that l μ = l μ. Acknowledements The second author is rateful to Professor B. Reznick of UIUC for discussions of the topic. The third author wants to acknowlede the hospitality of the Mathematics Department, Stockholm University in September-October The authors want to express their ratitude to the anonymous referee whose constructive criticism allowed us to substantially improve the quality of the exposition. The third author is supported by NSF rants DMS and DMS Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License ( which permits unrestricted use, distribution, and reproduction in any medium, provided you ive appropriate credit to the oriinal author(s) and the source, provide a link to the Creative Commons license, and indicate if chanes were made. References Arnol d, V.: The cohomoloy rin of the colored braid roup. Collect. Works 2, (2014) Beltrametti, M.C., Sommese, A.J.: The adjunction theory of complex projective varieties. de Gruyter Expositions in Mathematics. Walter de Gruyter & Co., Berlin (1995) Chiantini, L., Ciliberto, C.: On the dimension of secant varieties. J. Eur. Math. Soc. (JEMS) 12(5), (2010) Chiantini, L., Ottaviani, G., Vannieuwenhoven, N.: On eneric identifiability of symmetric tensors of subeneric rank, arxiv: v2, April (2015) Eisenbud, D., Harris, J.: On varieties of minimal deree (a centennial account). Alebraic eometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 3 13, In: Proceedins of Symposia in Pure Mathematics, 46, Part 1, American Mathematical Society, Providence, RI, (1987) Khesin, B., Shapiro, B.: Swallowtails and Whitney umbrellas are homeomorphic. J. Al. Geom. 1, (1992) Newman, D.J., Slater, M.: Warin s problem for the rin of polynomials. J. Number Theory 11(4), (1979) Reznick, B.: Linearly dependent powers of quadratic forms, Preliminary report: , slides. faculty.math.illinois.edu/~reznick/11111.pdf. Accessed 2 Dec 2017 Vassiliev, V.: Complements of discriminants of smooth maps: topoloy and applications. Translated from the Russian by B. Goldfarb. Translations of mathematical monoraphs, 98. American Mathematical Society, Providence, RI, vi+208 pp(1992)