Abstract
A linearized polynomial over \({{\mathbb {F}}}_{q^n}\) is called scattered when for any \(t,x\in {{\mathbb {F}}}_{q^n}\), the condition \(xf(t)-tf(x)=0\) holds if and only if x and t are \({\mathbb {F}}_q\)-linearly dependent. General conditions for linearized polynomials over \({{\mathbb {F}}}_{q^n}\) to be scattered can be deduced from the recent results in Csajbók (Scalar q-subresultants and Dickson matrices, 2018), Csajbók et al. (Finite Fields Appl 56:109–130, 2019), McGuire and Sheekey (Finite Fields Appl 57:68–91, 2019), Polverino and Zullo (On the number of roots of some linearized polynomials, 2019). Some of them are based on the Dickson matrix associated with a linearized polynomial. Here a new condition involving Dickson matrices is stated. This condition is then applied to the Lunardon–Polverino binomial \(x^{q^s}+\delta x^{q^{n-s}}\), allowing to prove that for any n and s, if \({{\,\mathrm{N}\,}}_{q^n/q}(\delta )=1\), then the binomial is not scattered. Also, a necessary and sufficient condition for \(x^{q^s}+bx^{q^{2s}}\) to be scattered is shown which is stated in terms of a special plane algebraic curve.
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1 Introduction
A point P of the projective space \(\mathrm {PG}(d-1,q^n)\) is a one-dimensional subspace of the vector space \({\mathbb {F}}_{q^n}^d\); that is, \(P=\langle v\rangle _{{\mathbb {F}}_{q^n}}=\{cv:c\in {\mathbb {F}}_{q^n}\}\) for some nonzero \(v\in {\mathbb {F}}_{q^n}^d\).
Let U be an r-dimensional \({\mathbb {F}}_q\)-subspace of \({\mathbb {F}}_{q^n}^d\). Then
is an \({\mathbb {F}}_q\)-linear set (or just linear set) of rank r in \(\mathrm {PG}(d-1,q^n)\). Let \(u,v\in U\). If \(u=cv\), \(c\in {\mathbb {F}}_q\), then clearly \(\langle u\rangle _{{\mathbb {F}}_{q^n}}=\langle v\rangle _{{\mathbb {F}}_{q^n}}\). If this is the only case in which two vectors of U determine the same point of \(\mathrm {PG}(d-1,q)\), that is, \(\langle v\rangle _{{\mathbb {F}}_{q^n}}=\langle u\rangle _{{\mathbb {F}}_{q^n}}\) if and only if \(\langle v\rangle _{{\mathbb {F}}_q}=\langle u\rangle _{{\mathbb {F}}_q}\), then \(L_U\) is called a scattered linear set. Equivalently, \(L_U\) is scattered if and only if it has maximum size \((q^r-1)/(q-1)\) with respect to r. The linear sets are related to combinatorial objects, such as blocking sets, two-intersection sets, finite semifields, rank-distance codes, and many others. The interested reader is referred to the survey by Polverino [18] and to [20], where J. Sheekey builds a bridge with the rank-distance codes.
Assume that in particular U is an \({\mathbb {F}}_q\)-subspace of \({\mathbb {F}}_{q^n}^2\), \(\dim _{{\mathbb {F}}_q}U=n\). In this case \(L_U:=\{\langle v\rangle _{{\mathbb {F}}_{q^n}}:v\in U,v\ne 0\}\subseteq \mathrm {PG}(1,q^n)\), is called a maximum linear set of \(\mathrm {PG}(1,q^n)\), since by the dimension formula any linear set of rank greater than n equals \(\mathrm {PG}(1,q^n)\). Up to projectivities of \(\mathrm {PG}(1,q^n)\) it may be assumed that \(\langle (0,1)\rangle _{{\mathbb {F}}_{q^n}}\not \in L_U\). Hence
where f(x) is a suitable \({\mathbb {F}}_q\)-linear map, that is a linearized polynomial:
If \(L_f\) is scattered, then f(x) is called a scattered linearized polynomial, or scattered q-polynomial with respect to n. A property characterizing the scattered q-polynomials is that for any \(x,y\in {\mathbb {F}}_{q^n}^*\), \({f(x)}/x={f(y)}/y\) if and only if \(\langle x\rangle _{{\mathbb {F}}_q}=\langle y\rangle _{{\mathbb {F}}_q}\).
A first example of scattered q-polynomial is \(f(x)=x^q\) [3], with respect to any n. Indeed, for any \(x,y\in {\mathbb {F}}_{q^n}^*\), \({f(x)}/x={f(y)}/y\), is equivalent to \(x^{q-1}=y^{q-1}\), hence to \(x/y\in {\mathbb {F}}_q^*\). A derived example is \(f(x)=x^{q^s}\), \(\gcd (n,s)=1\). Indeed \((x/y)^{q^s-1}=1\) implies \(x/y\in {\mathbb {F}}_{q^s}\cap {\mathbb {F}}_{q^n}^*={\mathbb {F}}_q^*\). In both cases above, \(L_f=\{\langle (x,f(x))\rangle _{{\mathbb {F}}_{q^n}}:x\in {\mathbb {F}}_{q^n}^*\}= \{\langle (1,z)\rangle _{{\mathbb {F}}_{q^n}}:z\in {\mathbb {F}}_{q^n},\,{{\,\mathrm{N}\,}}_{q^n/q}(z)=1\}\), where \({{\,\mathrm{N}\,}}_{q^n/q}(z)=z^{(q^n-1)/(q-1)}\) denotes the norm over \({\mathbb {F}}_q\) of \(z\in {\mathbb {F}}_{q^n}\). The related linear set is called a linear set of pseudoregulus type.
The next example has been given by Lunardon and Polverino [12] and generalized in [11, 20]:
In particular cases, the condition \({{\,\mathrm{N}\,}}_{q^n/q}(\delta )\ne 1\) has been proved to be necessary for f(x) to be scattered [2, 10, 11, 22]. In Sect. 3 it will proved that actually it is necessary for any n and s. Further examples of scattered q-polynomials are given in [5, 6, 14, 22]. All of them are with respect to \(n=6\) or \(n=8\). Bartoli et al. [1] proved that if \({\hat{f}}(x)\) is the adjoint of f(x) with respect to the bilinear form \(\langle x,y\rangle =\hbox {Tr}_{q^n/q}(xy)\) in \({\mathbb {F}}_{q^n}^2\), where \(\hbox {Tr}_{q^n/q}(z)=\sum _{i=0}^{n-1}z^{q^i}\) denotes the trace over \({\mathbb {F}}_q\) of \(z\in {\mathbb {F}}_{q^n}\), then \(L_f=L_{{\hat{f}}}\). This implies that if the polynomial f(x) in (1) is scattered, then also \({\hat{f}}(x)=\sum _{i=0}^{n-1}a_i^{q^{n-i}}x^{q^{n-i}}\) is. Up to the knowledge of the author of this paper, no more examples of scattered q-polynomials are known. So, it would seem that scattered q-polynomials are rare. Bartoli and Zhou [2] formalized such an idea of scarcity by proving that the pseudoregulus and Lunardon–Polverino polynomials are, roughly speaking, the only q-polynomials of a certain type which are scattered for infinitely many n.
Recently, a great deal of effort has been put in finding conditions for q-polynomials to be scattered [4, 7, 15, 19]. Some of them are based on the Dickson matrix associated with the q-polynomial in (1), that is, the \(n\times n\) matrix
It is well-known that the rank of \(M_{q,f}\) equals the rank of f(x), see for example [21, Proposition 4.4]. This rank can be computed by applying the following result by B. Csajbók:
Theorem 1.1
([4, Theorem 3.4]) Let \(M_{q,f}\) be the Dickson matrix associated with the q-polynomial in (1). Denote by \(M_{q,f}^{(r)}\) the \(r\times r\) submatrix of \(M_{q,f}\) obtained by considering the last r columns and the first r rows of \(M_{q,f}\). Then the rank of f(x) is t if and only if \(\det (M_{q,f}^{(n)}) =\det (M_{q,f}^{(n-1)}) =\cdots = \det (M_{q,f}^{(t+1)}) =0\), and \(\det (M_{q,f}^{(t)}) \ne 0\).
A q-polynomial \(f(x)\in {\mathbb {F}}_{q^n}[x]\) is scattered if and only if for any \(m\in {\mathbb {F}}_{q^n}\) the dimension of the kernel of \(f_m(x)=mx+f(x)\) is at most one. So, by Theorem 1.1 a necessary and sufficient condition for f(x) to be scattered is that the system of two equations
has no solution in the variable \(m\in {\mathbb {F}}_{q^n}\).
In this paper a condition consisting of one equation (Proposition 2.2) is proved, and applied to two binomials. It would seem that one equation is better than two in order to prove that a given q-polynomial f(x) is not scattered, while two equations will usually be more helpful in the proof that f(x) is. As a matter of fact, here the condition \({{\,\mathrm{N}\,}}_{q^n/q}(\delta )\ne 1\) is proved to be necessary for the Lunardon–Polverino binomial to be scattered (cf. Theorem 3.4). Furthermore, two necessary and sufficient conditions for \(x^{q^s}+bx^{q^{2s}}\) (where \(\gcd (s,n)=1\)) to be scattered are stated in Propositions 3.5 and 3.10 . This leads to the fact that the polynomial \(x^q+bx^{q^2}\), \(b\ne 0\), is never scattered if \(n\ge 5\) (cf. Proposition 3.8 and Remark 3.11).
2 A condition for scattered linearized polynomials
In this paper s, n, q and \(\sigma \) will always denote natural numbers such that \(n\ge 3\), \(\gcd (s,n)=1\), q is the power of a prime and \(\sigma =q^s\). Any \({\mathbb {F}}_q\)-linear endomorphism of \({\mathbb {F}}_{q^n}\) can be represented in the form
As a matter of fact, if \(\tau \) is the permutation \(i\mapsto is\) of \({\mathbb {Z}}/(n)\), then f(x) is the same function of \(\tilde{f}(x)=\sum _{i=0}^{n-1}a_{\tau ^{-1}(i)}x^{q^i}\). Generalizing the notion of Dickson matrix given in the previous section, the \(\sigma \)-matrix of Dickson associated with the linearized polynomial \(g(t)=\sum _{i=0}^{n-1}a_it^{\sigma ^i}\) is
This is just the Dickson matrix \(M_{q,{\tilde{g}}}\) associated with \({\tilde{g}}(t)\) after a permutation of the row and columns. Indeed, the element in row r and column c of \(M_{q,{\tilde{g}}}\), \(r,c\in \{0,1,\ldots ,n-1\}\), is \(m_{rc}=a_{\tau ^{-1}(c-r)}^{q^r}=a_{\tau ^{-1}(c)-\tau ^{-1}(r)}^{q^r}\). By applying \(\tau \) to both the row and column index, \(m_{\tau (r)\tau (c)}=a_{c-r}^{\sigma ^r}\) follows. Therefore, the rank of \(M_{\sigma ,g}\) equals the rank of g(t).
Remark 2.1
Each row of an n-order \(\sigma \)-matrix of Dickson is obtained from the previous one (cyclically) by the map
which is an invertible semilinear map of \({{\mathbb {F}}}_{q^n}^n\) into itself.
The polynomial (2) is scattered if and only if \(f_1(x)=\sum _{i=1}^{n-1}a_ix^{\sigma ^i}\) is. Hence in the following \(a_0\) will always be zero.
Proposition 2.2
Let \(f(x)=\sum _{i=1}^{n-1}a_ix^{\sigma ^i}\) be a linearized polynomial over \({{\mathbb {F}}}_{q^n}\), and
Then the following conditions are equivalent:
- (i)
the polynomial f(x) is scattered;
- (ii)
for any \(x\in {{\mathbb {F}}}_{q^n}^*\), a nonsingular \((n-1)\)-order minor of \(M_{\sigma ,g}\) exists;
- (iii)
for any \(x\in {{\mathbb {F}}}_{q^n}^*\), all \((n-1)\)-order minors of \(M_{\sigma ,g}\) are nonsingular.
Proof
The polynomial f(x) is scattered if and only if for any \(x\in {{\mathbb {F}}}_{q^n}^*\) the rank of \(h(t)=xf(t)-tf(x)\) is \(n-1\), that is, the rank of
is always \(n-1\). By dividing the rows of \(M_{\sigma ,h}\) by \(x{\hbox {'}}s\), \(x^\sigma \), \(x^{\sigma ^2}\), \(\ldots \), \(x^{\sigma ^{n-1}}\), respectively, and then multiplying the columns for that same elements, one obtains
that is, the matrix \(M_{\sigma ,g}\). By Remark 2.1, if a \(\sigma \)-matrix of Dickson is singular, then any row is a linear combination of the remaining ones. Hence the rank of \(M_{\sigma ,g}\) equals the rank of any \((n-1)\times n\) matrix obtained from it by deleting a row. Furthermore, since the sum of the columns of \(M_{\sigma ,g}\) is zero, all \((n-1)\)-order minors have the same rank of \(M_{\sigma ,g}\). \(\square \)
3 Two linearized binomials
Definition 3.1
For any \(\delta \in {{\mathbb {F}}}_{q^n}\),
is the Lunardon–Polverino binomial.
If \({{\,\mathrm{N}\,}}_{q^n/q}(\delta )\ne 1\), then \(f_{\sigma ,\delta }\) is scattered [11,12,13, 20].
Proposition 3.2
The polynomial \(f_{\sigma ,\delta }(x)\) is scattered if only if there is no \(x\in {{\mathbb {F}}}_{q^n}^*\) such that
where \(z=\delta x^{\sigma ^{n-1}-\sigma }\).
Proof
The \((n-1)\)-th order North-West principal minor of the \(\sigma \)-matrix of Dickson associated with the polynomial
further normalized row by row, is
By Laplace expansion along the last column and induction on n, the determinant of B(z) can be computed as \((-1)^{n+1}\sum _{i=0}^{n-1}z^{(\sigma ^i-1)/(\sigma -1)}\). \(\square \)
The following can be useful in understanding the role of \(\delta \):
Proposition 3.3
Let \(z\in {\mathbb {F}}_{q^n}\). Then (3) holds if and only if there exists a \(y\in {\mathbb {F}}_{q^n}^*\) such that \(z=y^{\sigma -1}\) and \(\hbox {Tr}_{q^n/q}(y)=0\).
Proof
Any solution of (3) is nonzero. Raising \(\sum _{i=0}^{n-1}z^{(\sigma ^i-1)/(\sigma -1)}\) to the \(\sigma \), multiplying by z and then subtracting to the original equation yields \(1-{{\,\mathrm{N}\,}}_{q^n/q}(z)=0\). So, z is a solution of (3) if and only if \(z=y^{\sigma -1}\) for some \(y\in {\mathbb {F}}_{q^n}^*\), and \(\sum _{i=0}^{n-1}y^{\sigma ^i-1}=0\). The latter equation is equivalent to \(\hbox {Tr}_{q^n/q}(y)=0\). \(\square \)
Propositions 3.2 and 3.3 together show that, if \(f_{\sigma ,\delta }\) is not scattered, then there is an \(x\in {\mathbb {F}}_{q^n}\) such that \({{\,\mathrm{N}\,}}_{q^n/q}(\delta x^{\sigma ^{n-1}-\sigma })={{\,\mathrm{N}\,}}_{q^n/q}(\delta ) {{\,\mathrm{N}\,}}_{q^n/q}(x^{\sigma ^{n-1}-\sigma })=1\). On the other hand, \(q-1\) divides \(\sigma ^{n-1}-\sigma \), hence \({{\,\mathrm{N}\,}}_{q^n/q}( x^{\sigma ^{n-1}-\sigma })=1\). Summarizing, if \({{\,\mathrm{N}\,}}_{q^n/q}(\delta )\ne 1\), then the Lunardon–Polverino binomial is scattered, as is known.
Theorem 3.4
If \({{\,\mathrm{N}\,}}_{q^n/q}(\delta )=1\), then the Lunardon–Polverino binomial \(f_{\sigma ,\delta }(x)\) is not scattered.
Proof
\(\underline{\hbox {Case odd } n.}\) Since
and \(\gcd (s(n-2),n)=1\), the expression \(x^{\sigma ^{n-1}-\sigma }\) takes all values in \({{\mathbb {F}}}_{q^n}\) whose norm over \({\mathbb {F}}_q\) is equal to one. This allows the substitution \(\delta x^{\sigma ^{n-1}-\sigma }=w^{\sigma -1}\) into (3). So, \(f_{\sigma ,\delta }(x)\) is not scattered if and only if \(\hbox {Tr}_{q^n/q}(w)=0\) for some nonzero w and this is trivial.
\(\underline{\hbox {Case even }n.}\) Since \(\gcd (\sigma ^{n-2}-1,\sigma ^n-1)=\sigma ^2-1\), the set of all powers of elements in \({\mathbb {F}}_{q^n}\) with exponent \(\sigma ^{n-2}-1\) coincides with the set of all powers with exponent \(\sigma ^{2}-1\). Hence for any \(x\in {{\mathbb {F}}}_{q^n}\) there exists \(u\in {{\mathbb {F}}}_{q^n}\) such that \(x^{\sigma ^{n-1}-\sigma }=u^{\sigma ^2-1}\), and conversely. This allows the substitution \(z=\delta u^{\sigma ^2-1}\) in (3), meaning that if there is u such that
then \(f_{\sigma ,\delta }(x)\) is not scattered. So, taking \(\delta =d^{\sigma -1}\), (5) is equivalent to \(\hbox {Tr}_{q^n/q}(du^{\sigma +1})=0\), \(u\ne 0\). This is a quadratic form in u in a vector space over \({\mathbb {F}}_q\) of dimension greater than two which has at least one nontrivial zero.\(\square \)
The theorem above has been proved in the particular cases \(n=4\) in [10], \(s=1\) in [2], both n and q odd in [11], and odd n in [22].
Proposition 3.5
The polynomial \(f(x)=x^\sigma +bx^{\sigma ^2}\) is scattered if only if there is no \(x\in {{\mathbb {F}}}_{q^n}^*\) such that
Proof
The \(\sigma \)-matrix of Dickson associated with the polynomial
further normalized by dividing the rows by \(bx^{\sigma ^2}\), \(b^{\sigma }x^{\sigma ^3}\), \(\ldots \) is
The matrix obtained by deleting the last row and first column is B(w) (cf. (4)). \(\square \)
Corollary 3.6
Assume \(b_1,b_2\in {{\mathbb {F}}}_{q^n}\) and \({{\,\mathrm{N}\,}}_{q^n/q}(b_1)={{\,\mathrm{N}\,}}_{q^n/q}(b_2)\). Then the polynomials \(f_i(x)=x^\sigma +b_ix^{\sigma ^2}\), \(i=1,2\), are either both scattered, or both non-scattered.
Proof
If the norm of \(b_1\) is zero then the statement is trivial, so assume that it is not. Define \(w_i(x)=-(1+b_i^{-1}x^{\sigma -\sigma ^2})\) for \(i=1,2\), and note that \(w_1(x)=w_2(y)\) is equivalent to \(b_1/b_2=((x/y)^\sigma )^{\sigma -1}\), that is, \(((x/y)^\sigma )^{\sigma -1}=c^{\sigma -1}\) for some \(c\in {\mathbb {F}}_{q^n}^*\). This equation can be always solved in both x and y, whence \(w_1(x)\) and \(w_2(y)\) take the same set of values. \(\square \)
Remark 3.7
Corollary 3.6 allows to look at only \(q-1\) linearized polynomials, given s, n, and q. This makes a computer search easier. Computations with GAPFootnote 1 show that there are no scattered linearized polynomials of the form \(l_b(x)=x^q+bx^{q^2}\), \(b\ne 0\), for any \(q<223\) if \(n=5\). In [17] it is proved that for \(n=5\) and \(q\ge 223\) the linearized polynomial \(l_b(x)\) is not scattered for any \(b\ne 0\). The next proposition summarizes this.
Proposition 3.8
If \(n=5\) and \(b\in {{\mathbb {F}}}_{q^5}^*\), then the q-polynomial \(l_b(x)=x^q+bx^{q^2}\in {{\mathbb {F}}}_{q^5}[x]\) is non-scattered.
Remark 3.9
For \(n=4\) there are scattered polynomials of type \(l_b(x)\), \(b\ne 0\). By the results in [9, 10], all the related linear sets are of Lunardon–Polverino type, up to collineations.
Proposition 3.10
Let \(b\in {\mathbb {F}}_{q^n}^*\). The polynomial \(x^\sigma +bx^{\sigma ^2}\in {\mathbb {F}}_{q^n}[x]\) is not scattered if and only if the algebraic curve \(b^{-1}X^{q-1}+Y^{\sigma -1}+1=0\) in \(\mathrm {AG}(2,q^n)\) has a point \((x_0,y_0)\) with coordinates in \({\mathbb {F}}_{q^n}^*\), such that \(\hbox {Tr}_{q^n/q}(y_0)=0\).
Proof
By Proposition 3.3, the first equation in (6) is equivalent to the existence of \(y\in {\mathbb {F}}_{q^n}^*\) such that \(w=y^{\sigma -1}\), \(\hbox {Tr}_{q^n/q}(y)=0\). The second equation \(y^{\sigma -1}+1+b^{-1}x^{q-q^2}=0\) has solutions with \(x\ne 0\) if and only if \(b^{-1}x^{q-1}+y^{\sigma -1}+1=0\) does. \(\square \)
Remark 3.11
Very recently, Montanucci [16] proved that if \(n>5\), then for any q the algebraic curve \(b^{-1}X^{q-1}+Y^{q-1}+1=0\) has a point with the properties above. Together with Propositions 3.8 and 3.10 this implies that for \(n\ge 5\) no q-polynomial of type \(l_b(x)=x^q+bx^{q^2}\), \(b\ne 0\), is scattered.
Notes
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Zanella, C. A condition for scattered linearized polynomials involving Dickson matrices. J. Geom. 110, 50 (2019). https://doi.org/10.1007/s00022-019-0505-z
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DOI: https://doi.org/10.1007/s00022-019-0505-z