Key Polynomial Research Articles

Upper bounds of dual flagged Weyl characters

For a subset D of boxes in an n×n square grid, let χD(x) denote the dual character of the flagged Weyl module associated to D. It is known that χD(x) specifies to a Schubert polynomial (resp., a key polynomial) in the case when D is the Rothe diagram of a permutation (resp., the skyline diagram of a composition). One can naturally define a lower and an upper bound of χD(x). Mészáros, St. Dizier and Tanjaya conjectured that χD(x) attains the upper bound if and only if D avoids a certain single subdiagram. We provide a proof of this conjecture.

Key polynomials and preminimal pairs

The paper pertains to the theory of valued fields and is devoted to describing extensions of valuations to the field of rational functions in one variable. An efficient tool here is the notion of a key polynomial introduced by S. Maclane in 1936. With the use of the notion of a preminimal pair introduced recently, a new description of key polynomials over Henselian fields is found.

Algebraic classification of 2+1 geometries: a new approach

We present a convenient method of algebraic classification of 2+1 spacetimes into the types I, II, D, III, N and O, without using any field equations. It is based on the 2+1 analogue of the Newman–Penrose curvature scalars ΨA of distinct boost weights, which are specific projections of the Cotton tensor onto a suitable null triad. The algebraic types are then simply determined by the gradual vanishing of such Cotton scalars, starting with those of the highest boost weight. This classification is directly related to the specific multiplicity of the Cotton-aligned null directions and to the corresponding Bel–Debever criteria. Using a bivector (that is 2-form) decomposition, we demonstrate that our method is fully equivalent to the usual Petrov-type classification of 2+1 spacetimes based on the eigenvalue problem and determining the respective canonical Jordan form of the Cotton–York tensor. We also derive a simple synoptic algorithm of algebraic classification based on the key polynomial curvature invariants. To show the practical usefulness of our approach, we perform the classification of several explicit examples, namely the general class of Robinson–Trautman spacetimes with an aligned electromagnetic field and a cosmological constant, and other metrics of various algebraic types.

Monk's Rule for Demazure Characters of the General Linear Group

Key polynomials are characters of Demazure modules for the general linear group that generalize the Schur polynomials. We prove a nonsymmetric generalization of Monk's rule by giving a cancellation-free, multiplicity-free formula for the key polynomial expansion of the product of an arbitrary key polynomial with a degree one key polynomial.

Graded rings associated to valuations and direct limits

In this paper, we study the structure of the graded ring associated to a limit key polynomial Qn in terms of the key polynomials that define Qn. In order to do that, we use direct limits. In general, we describe the direct limit of a family of graded rings associated to a totally ordered set of valuations. As an example, we describe the graded ring associated to a valuation-algebraic valuation as a direct limit of graded rings associated to residue-transcendental valuations.

On truncations of valuations

In this paper we study the truncation νq of a valuation ν on a polynomial q. It is known that when q is a key polynomial, then νq is a valuation. It is also known that the converse does not hold. We show that when q is a key polynomial, then νq is the restriction of the truncation given by an optimizing root of q. We also discuss which conditions assure that νq=ν. Finally, we assume that νq is a valuation and present some conditions, given in terms of the graded algebra, to assure that q is a key polynomial.

Slide Multiplicity Free Key Polynomials

Schubert polynomials are refined by the key polynomials of Lascoux-Schützen-berger, which in turn are refined by the fundamental slide polynomials of Assaf-Searles. In this paper we determine which fundamental slide polynomial refinements of key polynomials, indexed by strong compositions, are multiplicity free. We also give a recursive algorithm to determine all terms in the fundamental slide polynomial refinement of a key polynomial indexed by a strong composition. From here, we apply our results to begin to classify which fundamental slide polynomial refinements, indexed by weak compositions, are multiplicity free. We completely resolve the cases when the weak composition has at most two nonzero parts or the sum has at most two nonzero terms.

Valuations on K[x] approaching a fixed irreducible polynomial

For a fixed irreducible polynomial F we study the set VF of all valuations on K[x] bounded by valuations whose support is (F). The first main result presents a characterization for valuations in VF in terms of their graded rings. We also present a result which gives, for a fixed ν∈VF and a key polynomial Q∈KP(ν), the maximum value that augmented valuations in VF can assume on Q. This value is presented explicitly in terms of the slopes of the Newton polygon of F with respect to Q. Finally, we present some results about Artin-Schreier extensions that illustrate the applications that we have in mind for the results in this paper.

Upper bounds of Schubert polynomials

Let w be a permutation of {1, 2, …, n}, and let D(w) be the Rothe diagram of w. The Schubert polynomial \({\mathfrak{S}_w}\left(x \right)\) can be realized as the dual character of the flagged Weyl module associated with D(w). This implies the following coefficient-wise inequality: $${\rm{Mi}}{{\rm{n}}_w}\left(x \right) \le {\mathfrak{S}_w}\left(x \right) \le {\rm{Ma}}{{\rm{x}}_w}\left(x \right),$$where both Minw(x) and Maxw(x) are polynomials determined by D(w). Fink et al. (2018) found that \({\mathfrak{S}_w}\left(x \right)\) equals the lower bound Minw(x) if and only if w avoids twelve permutation patterns. In this paper, we show that \({\mathfrak{S}_w}\left(x \right)\) reaches the upper bound Maxw(x) if and only if w avoids two permutation patterns 1432 and 1423. Similarly, for any given composition α ∈ ℤ n≽0 , one can define a lower bound Minα(x) and an upper bound Maxα(x) for the key polynomial κα(x). Hodges and Yong (2020) established that κα(x) equals Minα(x) if and only if α avoids five composition patterns. We show that κα(x) equals Maxα(x) if and only if α avoids a single composition pattern (0, 2). As an application, we obtain that when α avoids (0, 2), the key polynomial κα(x) is Lorentzian, partially verifying a conjecture of Huh et al. (2019).

On common extensions of valued fields

Given a valuation v on a field K, an extension v¯ to an algebraic closure K¯ and an extension w to K(X). In this paper, we study common extensions w¯ of both w and v¯ to the field K¯(X). We describe the set of valuations w¯ using a last key polynomial for the valuation w and its roots.

Limit key polynomials as p-polynomials

The main goal of this paper is to characterize limit key polynomials for a valuation ν on K[x]. We consider the set Ψα of key polynomials for ν of degree α. We set p to be the exponent characteristic of ν. Our first main result (Theorem 1.1) is that if F is a limit key polynomial for Ψα, then the degree of F is prα for some r∈N. Moreover, in Theorem 1.2, we show that there exist Q∈Ψα and F a limit key polynomial for Ψα, such that the Q-expansion of F only has terms which are powers of p.

Vertices of Schubitopes

Schubitopes were introduced by Monical, Tokcan and Yong as a specific family of generalized permutohedra. It was proven by Fink, Mészáros and St. Dizier that Schubitopes are the Newton polytopes of the dual characters of flagged Weyl modules. Important cases of Schubitopes include the Newton polytopes of Schubert polynomials and key polynomials. In this paper, we develop a combinatorial rule to generate the vertices of Schubitopes. As an application, we show that the vertices of the Newton polytope of a key polynomial can be generated by permutations in a lower interval in the Bruhat order, settling a conjecture of Monical, Tokcan and Yong.

Key polynomials over valued fields

Let K be a field. For any valuation µ on K[x] admitting key polynomials we determine the structure of the whole set of key polynomials in terms of a fixed key polynomial of minimal degree. We deduce a canonical bijection between the set of µ-equivalence classes of key polynomials and the maximal spectrum of the subring of elements of degree zero in the graded algebra of µ.

AES WITH CHAOTIC USING CHEBYSHEV POLYNOMIAL

All the important information is exchanged between facilities using the internet and networks, all these data should besecret and secured probably, the personal information of person in each of these institutions day by day need to organized secretlyand the need of the cryptography systems is raised which can easily encrypt the personal and critical data and it can be shared withother centers via internet without and concerns about privacy. Chaotic performance is added to different phases of AES but very few apply it on key generation and choosing ChebyshevPolynomial will provide a chaotic map which will led to random strong key. our system based on modified advanced encryptionstandard (AES) , with encryption and decryption in real time taking to consideration the criticality of data images that beenencrypted the main encryption algorithm is the same the modification is done by replacing the key generation algorithm byChebyshev Polynomial to generate key with the required key size.

Key polynomials and pseudo-convergent sequences

In this paper we introduce a new concept of key polynomials for a given valuation ν on K[x]. We prove that such polynomials have many of the expected properties of key polynomials as those defined by MacLane and Vaquié, for instance, that they are irreducible and that the truncation of ν associated to each key polynomial is a valuation. Moreover, we prove that every valuation ν on K[x] admits a sequence of key polynomials that completely determines ν (in the sense which we make precise in the paper). We also establish the relation between these key polynomials and pseudo-convergent sequences defined by Kaplansky.

Masking ring-LWE

In this paper, we propose a masking scheme to protect ring-LWE decryption from first-order side-channel attacks. In an unprotected ring-LWE decryption, the recovered plaintext is computed by first performing polynomial arithmetic on the secret key and then decoding the result. We mask the polynomial operations by arithmetically splitting the secret key polynomial into two random shares; the final decoding operation is performed using a new bespoke masked decoder. The outputs of our masked ring-LWE decryption are Boolean shares suitable for derivation of a symmetric key. Thus, the masking scheme keeps all intermediates, including the recovered plaintext, in the masked domain. We have implemented the masking scheme on both hardware and software. On a Xilinx Virtex-II FPGA, the masked ring-LWE processor requires around 2000 LUTs, a $$20~\%$$ increase in the area with respect to the unprotected architecture. A masked decryption operation takes 7478 cycles, which is only a factor $$2.6\times $$ larger than the unprotected decryption. On a 32-bit ARM Cortex-M4F processor, the masked software implementation costs around $$5.2\times $$ more cycles than the unprotected implementation.

Semistandard tableaux for Demazure characters (key polynomials) and their atoms

The Schur function indexed by a partition λ with at most n parts is the sum of the weight monomials for the Young tableaux of shape λ. Let π be an n-permutation. We give two descriptions of the tableaux that contribute their monomials to the key polynomial indexed by π and λ. (These polynomials are the characters of the Demazure modules for GL(n).) The “atom” indexed by π is the sum of weight monomials of the tableaux whose right keys are the “key” tableau for π. Schur functions and key polynomials can be decomposed into sums of atoms. We also describe the tableaux that contribute to an atom, the tableaux that have a left key equal to a given key, and the tableaux that have a left key bounded below by a given key.

Key polynomials

Let K→L be a field extension and ν a valuation of K. In order to study the possible extensions μ of ν to L (in the case of discrete ν of rank 1), S. MacLane introduced the notion of key polynomials, a certain well ordered subset of K[X], as well as the notion of augmented valuations, a way of building up μ by successive approximations. In a series of papers, M. Vaquié generalized MacLane’s notion of key polynomials to the case of arbitrary valuations ν (that is, valuations which are not necessarily discrete of rank 1). In the paper [3], Herrera Govantes, Olalla Acosta and Spivakovsky develop their own notion of key polynomials for extensions (K,ν)→(L,μ) of valued fields, where ν is of archimedean rank 1 (not necessarily discrete) and give an explicit description of the limit key polynomials. Our purpose in this paper is to clarify the relationship between the two notions of key polynomials already developed by Vaquié and F.J. Herrera Govantes, M.A. Olalla Acosta and M. Spivakovsky. Our main result is the comparison theorem (4.2). We also give an explicit example of a limit key polynomial in the case when ν and μ are centered in local noetherian rings.

Extension d’une valuation

We want to determine all the extensions of a valuation ν \nu of a field K K to a cyclic extension L L of K K , i.e. L = K ( x ) L=K(x) is the field of rational functions of x x or L = K ( θ ) L=K(\theta ) is the finite separable extension generated by a root θ \theta of an irreducible polynomial G ( x ) G(x) . In two articles from 1936, Saunders MacLane has introduced the notions of key polynomial and of augmented valuation for a given valuation μ \mu of K [ x ] K[x] , and has shown how we can recover any extension to L L of a discrete rank one valuation ν \nu of K K by a countable sequence of augmented valuations ( μ i ) i ∈ I \bigl (\mu _i\bigr ) _{i \in I} , with I ⊂ N I \subset \mathbb N . The valuation μ i \mu _i is defined by induction from the valuation μ i − 1 \mu _{i-1} , from a key polynomial ϕ i \phi _i and from the value γ i = μ ( ϕ i ) \gamma _i = \mu ( \phi _i ) . In this article we study some properties of the augmented valuations and we generalize the results of MacLane to the case of any valuation ν \nu of K K . For this we need to introduce simple admissible families of augmented valuations A = ( μ α ) α ∈ A {\mathcal A} = \bigl ( \mu _{\alpha } \bigr ) _{\alpha \in A} , where A A is not necessarily a countable set, and to define a limit key polynomial and limit augmented valuation for such families. Then, any extension μ \mu to L L of a valuation ν \nu on K K is again a limit of a family of augmented valuations. We also get a “factorization” theorem which gives a description of the values ( μ α ( f ) ) ( \mu _{\alpha } (f)) for any polynomial f f in K [ x ] K[x] .

Free Products in Prime Characteristic: A Representation of Fuchsian Groups

In 1988, S. White proved by means of field theory supplemented by a geometric argument that the real bijectionsx↦x+1 andx↦xd(dan odd prime) generate a free group of rank 2. When these maps are considered in prime characteristicp(so thatx↦x+1 generates a cyclic group of orderp) the geometric argument is no longer available. We show on the one hand that, generally, the geometry is redundant and on the other that, in characteristicp, further algebraic considerations are required to establish a key polynomial lemma. By these means we obtain an analogue of White's theorem for certain (countably) infinite subfieldsLof the algebraic closure of the finite prime field GF(p). For any (odd) primed, not a divisor ofp(p−1), the mapsx↦x+1 andx↦xdgenerate a group ofbijectionsof such a fieldLthat is isomorphic to the free product Z∗(Z/pZ). This implies an explicit natural algebraic faithful representation of the free product as a transitive permutation group on a countable set.