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Abstract: We start to study the problem of classifying smooth proper varieties over a field k from the standpoint of -homotopy theory. Motivated by the topological theory of surgery, we discuss the problem of classifying up to isomorphism all smooth proper varieties having a specified -homotopy type. Arithmetic considerations involving the sheaf of -connected components lead us to introduce several different notions of connectedness in -homotopy theory. We provide concrete links between these notions, connectedness of points by chains of affine lines, and various rationality properties of algebraic varieties (e.g., rational connectedness). We introduce the notion of an -h-cobordism, an algebro-geometric analog of the topological notion of h-cobordism, and use it as a tool to produce non-trivial -weak equivalences of smooth proper varieties. Also, we give explicit computations of refined -homotopy invariants, such as the -fundamental sheaf of groups, for some -connected varieties. We observe that the -fundamental sheaf of groups plays a central yet mysterious role in the structure of -h-cobordisms. As a consequence of these observations, we completely solve the classification problem for rational smooth proper surfaces over an algebraically closed field: while there exist arbitrary dimensional moduli of such surfaces, there are only countably many -homotopy types, each uniquely determined by the isomorphism class of its -fundamental sheaf of groups. [Copyright &y& Elsevier]

If X is an algebraic variety with at most canonical singularities and S is a ℚ-Cartier hypersurface in X, then the canonical threshold of the pair ( X, S) is defined as the least upper bound of the reals c for which the pair ( X, cS) is canonical. We show that the set of all possible canonical thresholds of the pairs ( X, S), where X is smooth and three-dimensional, satisfies the ascending chain condition. We also derive a formula for the canonical threshold of the pair (ℂ, S), where S is a Brieskorn singularity. [ABSTRACT FROM AUTHOR]

Abstract: Let be a smooth irreducible nondegenerate projective variety and let denote its dual variety. The locus of bitangent hyperplanes, i.e. hyperplanes tangent to at least two points of X, is a component of the singular locus of . In this paper we provide a sufficient condition for this component to be of maximal dimension and show how it can be used to determine which dual varieties of Grassmannians are normal. That last part may be compared to what has been done for hyperdeterminants by J. Weyman and A. Zelevinsky (1996) in . [Copyright &y& Elsevier]

Let X ⊂ (H0(L)) be a smooth projective variety embedded by the complete linear system associated to a very ample line bundle L on X. We call [image omitted] the section module of L. It has been known that the syzygies of RL as R = Sym(H0(L))-module play important roles in understanding geometric properties of X [2, 3, 5, 9, 10] even if X is not projectively normal. Generalizing the case of N2, p [2, 10], we prove some uniform theorems on higher normality and syzygies of a given linearly normal variety X and general inner projections when RL satisfies property N3, p (Theorems 1.1, 1.2, and Proposition 3.1). In particular, our uniform bounds are sharp as hyperelliptic curves and elementary transforms of elliptic ruled surfaces show. [ABSTRACT FROM AUTHOR]

Abstract: The intersection between a nilpotent orbit and the Lie algebra of a Borel subgroup is an equidimensional, quasi-affine algebraic variety. Its irreducible components are called orbital varieties. In this Note, we provide criteria to guarantee that an orbital variety is smooth or has a dense orbit for the adjoint action of B. In addition, we point out a possible relation between these two properties. [Copyright &y& Elsevier]

Abstract: For many codes defined geometrically over (e.g. coming from a finite complete intersection or a vector bundle on a projective variety) we prove the existence of an extension (no explicit lower bound for e) such that the associated code over has a strong uniformity property (all submatrices of a certain type have the same rank). [Copyright &y& Elsevier]

Generalized hypersubstitutions are mappings from the set of all fundamental operations into the set of all terms of the same language do not necessarily preserve the arities. Strong hyperidentities are iden- tities which are closed under the generalized hypersubstitutions and a strongly solid variety is a variety which every its identity is a strong hyperidentity. In this paper we give an example of pre-strongly solid varieties of commutative semigroups and determine the least and the greatest pre-strongly solid variety of commutative semigroups. [ABSTRACT FROM AUTHOR]

If G is a finitely generated group and A is an algebraic group, then RA(G) = Hom(G, A) is an algebraic variety. Define the "dimension sequence" of G over A as Pd(RA(G)) = (Nd(RA(G)), ..., N0(RA(G))), where Ni(RA(G)) is the number of irreducible components of RA(G) of dimension i (0 ≤ i ≤ d) and d = Dim(RA(G)). We use this invariant in the study of groups and deduce various results. For instance, we prove the following: Theorem A.Let w be a nontrivial word in the commutator subgroup ofFn = 〈x1, ..., xn〉, and letG = 〈x1, ..., xn; w = 1〉. IfRSL(2, ℂ)(G)is an irreducible variety andV-1 = {ρ | ρ ∈ RSL(2, ℂ)(Fn), ρ(w) = -I} ≠ ∅, thenPd(RSL(2, ℂ)(G)) ≠ Pd(RPSL(2, ℂ)(G)). Theorem B.Let w be a nontrivial word in the free group on{x1, ..., xn}with even exponent sum on each generator and exponent sum not equal to zero on at least one generator. SupposeG = 〈x1, ..., xn; w = 1〉. IfRSL(2, ℂ)(G)is an irreducible variety, thenPd(RSL(2, ℂ)(G)) ≠ Pd(RPSL(2, ℂ)(G)). We also show that if G = 〈x1, . ., xn, y; W = yp〉, where p ≥ 1 and W is a word in Fn = 〈x1, ..., xn〉, and A = PSL(2, ℂ), then Dim(RA(G)) = Max{3n, Dim(RA(G′)) +2 } ≤ 3n + 1 for G′ = 〈x1, ..., xn; W = 1〉. Another one of our results is that if G is a torus knot group with presentation 〈x, y; xp = yt〉 then Pd(RSL(2, ℂ)(G))≠Pd(RPSL(2, ℂ)(G)). [ABSTRACT FROM AUTHOR]

We consider real algebraic varieties that are the intersection of two real quadrics. For brevity, we refer to such varieties as real biquadrics. The rigid isotopy classes of real biquadrics have been described long ago. In the present paper, we find the rigid isotopy classes in which the biquadrics are GM-varieties. [ABSTRACT FROM AUTHOR]

Let K be an algebraically closed field of positive characteristic p, and G be a linear algebraic group over K. We give a user friendly proof of Nagata's theorem that every finite-dimensional rational representation of G is completely reducible if and only if the connected component G0 is a torus and p does not divide the index (G : G0). [ABSTRACT FROM AUTHOR]

Abstract: In this paper we are going to give an explicit description of the so-called ring of conditions of conics, i.e. of the homogenous space . This is achieved by first describing its equivariant version. [Copyright &y& Elsevier]

For F a $${\wp}$$-adic field together with a $${\wp}$$-adic valuation, we present a new characterization for a map $${p: F^{n} \rightarrow {\mathbb R}\cup\{-\infty}\}$$ to be a $${\wp}$$- adic norm on the vector space F. This characterization was motivated by the concept of tight maps - maps that naturally arise within the theory of valuated matroids and tight spans. As an immediate consequence, we show that the two descriptions of the affine building of SL( F) in terms of (i) $${\wp}$$-adic norms given by Bruhat and Tits and (ii) tight maps given by Terhalle essentially coincide. The result suggests that similar characterizations of affine buildings of other classical groups should exist, and that the theory of affine buildings may turn out as a particular case of a yet to be developed geometric theory of valuated (and δ-valuated) matroids and their tight spans providing simply-connected G-spaces for large classes of appropriately specified groups G that could serve as a basis for an affine variant of Gromov's theory. [ABSTRACT FROM AUTHOR]

Let G be a connected reductive linear algebraic group defined over an algebraically closed field k of positive characteristic. Let Z(G) ⊂ G be the center, and G/Z(G) = \prod_{i=1}^m G_i$, where each Gi is simple with trivial center. For i ∈ [1, m], let ρi : G → Gi be the natural projection. Fix a proper parabolic subgroup P of G such that for each i ∈ [1, m], the image ρi(G) ⊂ Gi is a proper parabolic subgroup. Fix a strictly anti-dominant character χ of P such that χ is trivial on Z(G). Let M be a smooth projective variety, defined over k, equipped with a very ample line bundle ξ. Let EG → M be a principal G-bundle. We prove that the following six statements are equivalent: (1) The line bundle EG(χ) → EG/P associated to the principal P-bundle EG → EG/P for the character χ is numerically effective. (2) The sequence of principal G-bundles $\{(F^a_M)^*E_G\}_{a\geq 0}$ is bounded, where FM is the absolute Frobenius morphism of M. (3) The principal G-bundle EG is strongly semistable with respect to ξ, and c2(ad(EG)) is numerically equivalent to zero. (4) The principal G-bundle EG is strongly semistable with respect to ξ, and [c2(ad(EG))c1(ξ)d-2] = 0. (5) The adjoint vector bundle ad(EG) is numerically effective. (6) For every pair of the form (Y,ψ), where Y is an irreducible smooth projective curve and ψ : Y → M is a morphism, the principal G-bundle ψ*EG → Y is semistable. [ABSTRACT FROM AUTHOR]

We analyze several families of two-dimensional quantum random walks. The feasible region (the region where probabilities do not decay exponentially with time) grows linearly with time, as is the case with one-dimensional QRW. The limiting shape of the feasible region is, however, quite different. The limit region turns out to be an algebraic set, which we characterize as the rational image of a compact algebraic variety. We also compute the probability profile within the limit region, which is essentially a negative power of the Gaussian curvature of the same algebraic variety. Our methods are based on analysis of the space-time generating function, following the methods of Pemantle and Wilson (J. Comb. Theory, Ser. A 97(1):129-161, ). [ABSTRACT FROM AUTHOR]

Abstract: The totality of all Lie algebra structures on a vector space V over a field is an algebraic variety over on which the group acts naturally. We give an explicit description of for which is based on the notion of compatibility of Lie algebra structures. [ABSTRACT FROM AUTHOR]

Abstract: We develop the algebraic polynomial theory for “supertropical algebra,” as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of “ghost elements,” which also play the key role in our structure theory. Here, we work somewhat more generally over an ordered monoid, and develop a theory which contains the analogs of several basic theorems of classical commutative algebra. This structure enables one to develop a Zariski-type algebraic geometric approach to tropical geometry, viewing tropical varieties as sets of roots of (supertropical) polynomials, leading to an analog of the Hilbert Nullstellensatz. Particular attention is paid to factorization of polynomials. In one indeterminate, any polynomial can be factored into linear and quadratic factors, and although unique factorization may fail, there is a “preferred” factorization that is explained both geometrically and algebraically. The failure of unique factorization in several indeterminates is explained by geometric phenomena described in the paper. [ABSTRACT FROM AUTHOR]

Let G be a linear algebraic group over ℂ and P be a parabolic subgroup. We determine the signatures of the flag manifold G/P. As an application, we prove that the nonsingular hypersurfaces of degree 2 in ℂ P n are prime if n satisfies certain conditions. [ABSTRACT FROM AUTHOR]

Let V be a projective algebraic variety of degree D and dimension n nonsingular in codimension one. Then the construction of the normalization of V can be canonically reduced, within time polynomial in the size of the input and $$ {D^{{n^{O(1)}}}} $$, to solving a linear equation aX + bY + cZ = 0 over a polynomial ring. We describe a plan of proving this result with all lemmas. Bibliography: 4 titles. [ABSTRACT FROM AUTHOR]

Hanna Neumann asked whether it was possible for two non-isomorphic residually nilpotent finitely generated (fg) groups, one of them free, to share the lower central sequence. Baumslag answered the question in the affirmative and thus gave rise to parafree groups. A group G is termed parafree of rank n if it is residually nilpotent and shares the same lower central sequence with a free group of rank n. The deviation of a fg parafree group G of rank n is the difference μ(G) - n, where μ(G) is the minimum possible number of generators of G. Let G be fg; then Hom(G, SL2) inherits the structure of an algebraic variety, denoted by R(G). If G is an n generated parafree group, then the deviation of G is 0 iff Dim(R(G)) = 3n. It is known that for n ≥ 2 there exist infinitely many parafree groups of rank n and deviation 1 with non-isomorphic representation varieties of dimension 3n. In this paper it is shown that given integers n ≥ 2 and k ≥ 1, there exists infinitely many parafree groups of rank n and deviation k with non-isomorphic representation varieties of dimension different from 3n; in particular, there exist infinitely many parafree groups G of rank n with Dim(R(G)) > q, where q ≥ 3n is an arbitrary integer. [ABSTRACT FROM AUTHOR]

Based on the basis theorem of Bruhat–Chevalley (in Algebraic Groups and Their Generalizations: Classical Methods, Proceedings of Symposia in Pure Mathematics, vol. 56 (part 1), pp. 1–26, AMS, Providence, ) and the formula for multiplying Schubert classes obtained in (Duan, Invent. Math. 159:407–436, ) and programmed in (Duan and Zhao, Int. J. Algebra Comput. 16:1197–1210, ), we introduce a new method for computing the Chow rings of flag varieties (resp. the integral cohomology of homogeneous spaces). The method and results of this paper have been extended in (Duan and Zhao, and ) to obtain the integral cohomology rings of all complete flag manifolds, and to construct the integral cohomologies of Lie groups in terms of Schubert classes. [ABSTRACT FROM AUTHOR]

flat unar is a semilattice of height one with an additional unary operation. We characterise the primal, semi-primal and dualisable finite flat unars. Along the way, we give a useful characterisation of primality in general and use an array of techniques from the theory of natural dualities. [ABSTRACT FROM AUTHOR]

Abstract: Regular semisimple Hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory, numerical analysis, representation theory, algebraic geometry, and combinatorics. We give a “Giambelli formula” expressing the classes of regular semisimple Hessenberg varieties in terms of Chern classes. In fact, we show that the cohomology class of each regular semisimple Hessenberg variety is the specialization of a certain double Schubert polynomial, giving a natural geometric interpretation to such specializations. We also decompose such classes in terms of the Schubert basis for the cohomology ring of the flag variety. The coefficients obtained are nonnegative, and we give closed combinatorial formulas for the coefficients in many cases. We introduce a closely related family of schemes called regular nilpotent Hessenberg schemes, and use our results to determine when such schemes are reduced. [Copyright &y& Elsevier]

Let G/P be a rational homogeneous variety, where P is a parabolic subgroup of a simple and simply connected linear algebraic group G defined over an algebraically closed field of characteristic zero. A homogeneous principal bundle over G/P is semistable (respectively, polystable) if and only if it is equivariantly semistable (respectively, equivariantly polystable). A stable homogeneous principal H-bundle ( EH , ρ) is equivariantly stable, but the converse is not true in general. If a homogeneous principal H-bundle ( EH , ρ) is equivariantly stable, but the principal H-bundle EH is not stable, then the principal H-bundle EH admits an action ρ′ of G such that the pair ( EH , ρ′) is a homogeneous principal H-bundle which is not equivariantly stable. [ABSTRACT FROM AUTHOR]

We deal with homogeneous toric bundles M over generalized flag manifolds GC/ P, where G is a compact semisimple Lie group and P a parabolic subgroup. Using symplectic data, we provide a simple characterization of the homogeneous toric bundles M which are Fano; we then show that a homogeneous toric bundle M admits a Kähler-Ricci solitonic metric if and only if it is Fano. [ABSTRACT FROM AUTHOR]

The article presents a study which demonstrates that pseudoprimes on elliptic curves behave in many ways very similar to pseudoprimes related to Lucas sequences. It mentions that the author derives criteria for the group structure of complex multiplication (CM) curves, which allows for the testing for all composites. Also presented are several formulas that answer questions about certain types of elliptic pseudoprimes.

The parameter space $ \mathcal{S}_p$ is a smooth affine algebraic curve whose genus increases rapidly with $ p$ $ \mathcal{S}_p$ $ \mathcal{S}_p$ $ \mathcal{S}_p$ [ABSTRACT FROM AUTHOR]

We use the degree formula for connective K-theory to study rational contractions of algebraic varieties. As an application we obtain a condition of rational incompressibility of algebraic varieties and a version of the index reduction formula. Examples include complete intersection, rationally connected varieties, twisted forms of abelian varieties and Calabi-Yau varieties [ABSTRACT FROM AUTHOR]

In Boyer [Monodromy of perverse sheaves on vanishing cycles on some Shimura varieties, Invent. Math. 177 (2009), 239-280 (in French)], a sheaf version of the monodromy-weight conjecture for some unitary Shimura varieties was proved by giving explicitly the monodromy filtration of the complex of vanishing cycles in terms of local systems introduced in Harris and Taylor [The geometry and cohomology of some simple Shimura varieties (Princeton University Press, Princeton, NJ, 2001)]. The main result of this paper is the cohomological version of the monodromy-weight conjecture for these Shimura varieties, which we prove by means of an explicit description of the groups of cohomology in terms of automorphic representations and the local Langlands correspondence. [ABSTRACT FROM AUTHOR]

Using the Gille--Merkurjev norm principle we compute in a uniform way the image of the degree map for quadrics (Springer's theorem), for twisted forms of maximal orthogonal Grassmannians (a theorem of Bayer-Fluckiger and Lenstra), and for E6- (a theorem of Rost) and E7-varieties. [ABSTRACT FROM AUTHOR]

We classify the irreducible complex characters of the symplectic groups Sp2n(q) and the orthogonal groups [image omitted], Spin2n+1(q) of degrees up to the bound D, where D = (qn - 1)q4n-10/2 for symplectic groups, D = q4n-8 for orthogonal groups in odd dimension, and D = q4n-10 for orthogonal groups in even dimension. [ABSTRACT FROM AUTHOR]

Abstract: It is known that a tilting generator on an algebraic variety X gives a derived equivalence between X and a certain non-commutative algebra. In this paper, we present a method to construct a tilting generator from an ample line bundle, and construct it in several examples. [Copyright &y& Elsevier]

Abstract  We have developed in the past several algorithms with intrinsic complexity bounds for the problem of point finding in real algebraic varieties. Our aim here is to give a comprehensive presentation of the geometrical tools which are necessary to prove the correctness and complexity estimates of these algorithms. Our results form also the geometrical main ingredients for the computational treatment of singular hypersurfaces. In particular, we show the non–emptiness of suitable generic dual polar varieties of (possibly singular) real varieties, show that generic polar varieties may become singular at smooth points of the original variety and exhibit a sufficient criterion when this is not the case. Further, we introduce the new concept of meagerly generic polar varieties and give a degree estimate for them in terms of the degrees of generic polar varieties. The statements are illustrated by examples and a computer experiment. [ABSTRACT FROM AUTHOR]

We represent and analyze the general solution of the sixth Painlevé transcendent $$ \mathcal{P}_6 $$ in the Picard-Hitchin-Okamoto class in the Painlevé form as the logarithmic derivative of the ratio of τ-functions. We express these functions explicitly in terms of the elliptic Legendre integrals and Jacobi theta functions, for which we write the general differentiation rules. We also establish a relation between the $$ \mathcal{P}_6 $$ equation and the uniformization of algebraic curves and present examples. [ABSTRACT FROM AUTHOR]

Abstract: We show that the secant varieties of rank three compact Hermitian symmetric spaces in their minimal homogeneous embeddings are normal, with rational singularities. We show that their ideals are generated in degree three—with one exception, the secant variety of the 21-dimensional spinor variety in where we show that the ideal is generated in degree four. We also discuss the coordinate rings of secant varieties of compact Hermitian symmetric spaces. [Copyright &y& Elsevier]

We present a decomposition of the space of twisted arcs of a toric stack. As a consequence, we give a combinatorial description of the motivic integral associated to a torus-invariant divisor of a toric stack. [ABSTRACT FROM AUTHOR]

The real closed valuation rings, i.e., convex subrings of real closed fields, form a proper subclass of the class of real closed domains. It is shown how one can recognize whether a real closed domain is a valuation ring. This leads to a characterization of the totally ordered domains whose real closure is a valuation ring. Real closures of totally ordered factor rings of coordinate rings of real algebraic varieties are very frequently valuation rings. In particular, the real closure of the coordinate ring of a curve is an SV-ring (i.e., the factor rings modulo prime ideals are valuation rings). Real closed valuation rings play a role in the definition of real closed rings, as well as in the construction of real closures of rings and porings. They can also be used for the study of univariate differentiable semi-algebraic functions. This leads to the notion of differentiablility of semi-algebraic functions along half branches of curves. [ABSTRACT FROM AUTHOR]

The following two questions from univariate (curve) subdivision are not related to each other, but are both of current interest. (1) Is it possible for a non- stationary subdivision scheme to have as its limit an algebraic curve of genus higher than zero? (2) Is it possible to exploit the structure of a (stationary) univariate subdivision matrix to express the unit row eigenvector in closed form? My guess is that both answers are ‘No’, but until proofs of this are articulated it seems worth asking the questions. This note is offered in the hope that a reader will be able to provide some insight. [ABSTRACT FROM AUTHOR]

The article presents a family of quasiprojective surfaces, in which the Hilbert scheme is a nonsingular irreducible quasiprojective algebraic variety of dimension 2m. It examines the classical cohomology of the said varieties and describes the representation theory of Heisenberg algebras. It aims to study the small quantum product on the T-equivariant cohomology of Hilbert scheme.

In this paper, we present a new proof of the H-coloring dichotomy, which was first proved by Hell and Nešetřil in 1990, and then was reproved by Bulatov in 2005. Our proof is much shorter than the original proof and avoids the algebraic machinery of Bulatov's proof. [ABSTRACT FROM AUTHOR]

Abstract  Spectral properties of normal (2k+1)-banded Toeplitz matrices of order n, with k≤⌊ n/2⌋, are described. Formulas for the distance of (2k+1)-banded Toeplitz matrices to the algebraic variety of similarly structured normal matrices are presented. [ABSTRACT FROM AUTHOR]

Let G denote a finite group and π : Z → Y a Galois covering of smooth projective curves with Galois group G. For every subgroup H of G there is a canonical action of the corresponding Hecke algebra ℚ[ H\ G/ H] on the Jacobian of the curve X = Z/ H. To each rational irreducible representation  of G we associate an idempotent in the Hecke algebra, which induces a correspondence of the curve X and thus an abelian subvariety P of the Jacobian JX. We give sufficient conditions on , H, and the action of G on Z for P to be a Prym-Tyurin variety. We obtain many new families of Prym-Tyurin varieties of arbitrary exponent in this way. [ABSTRACT FROM AUTHOR]

We describe periodic groups saturated by groups in the set ℭ = { L2( r), r ⩾ 4; L3(2 m), m ⩾ 1; U3(2 m), m ⩾ 2; Sz(2 m), m ⩾ 3}. As a corollary we give a description of periodic groups G saturated by finite simple groups and satisfying one of the following conditions: (a) centralizers of involutions in G are 2-closed; (b) G contains a strongly embedded 2-local subgroup. [ABSTRACT FROM AUTHOR]

Given two affine algebraic varieties X and Y such that X ×  ≅ Y × , it may happen that X and Y are not isomorphic. This is shown, for example, by so-called “Danielewski surfaces.” In this note, we give a simple construction of such varieties X, Y of an arbitrary dimension n > 1. [ABSTRACT FROM AUTHOR]

Abstract: For a very ample line bundle L on a projective scheme X, let , , be the embedding defined by the complete linear series . In this paper we study the problem how the Castelnuovo–Mumford regularity of effects on the defining equations of and the syzygies among them. We show that if is m-regular, then satisfies property for , and satisfies property for all . [Copyright &y& Elsevier]

Abstract: Let X be a complex nonsingular projective 3-fold of general type. We show that there are positive constants c, and such that and for all . [Copyright &y& Elsevier]

Abstract: Starting from the well-known transformation law for the Klein functions, we give a proof of a fairly general multiplicative distribution formula for the Siegel functions associated to isogenous complex lattices. This formula has as an immediate consequence the remarkable distribution formula proved by Jarvis in 2000 on the occasion of Rolshausen''s thesis on the second K-group of an elliptic curve. [Copyright &y& Elsevier]

Abstract: We construct full strong exceptional collections of line bundles on smooth toric Fano Deligne–Mumford stacks of Picard number at most two and of any Picard number in dimension two. It is hoped that the approach of this paper will eventually lead to the proof of the existence of such collections on all smooth toric nef-Fano Deligne–Mumford stacks. [Copyright &y& Elsevier]

Abstract: We define push-forwards along projective morphisms in the Witt theory of smooth quasi-projective varieties over a field. We prove that they have standard properties such as functoriality, compatibility with pull-backs and projection formulas. [Copyright &y& Elsevier]

The article focuses on the concept of the mathematics taught in the classroom as well as the conceptual richness of systems of equations. According to the authors, the solving systems of equations is often introduced through the use of graphs, then the focus is turned to completing exercises as well as solving problems by using several algebraic methods. They also queries how the teaching systems of equations can be enhanced to allow students to study algebraic manipulations and problems that involve linear equations from a more comprehensive perspective.

The article provides information on the inequalities of lattice polygons in relation to Scott's inequality and the equation 2i + 7. It states that algebraic geometry is used to study lattice equivalent polygons, since toric geometry is a powerful link that connects discrete and algebraic geometry. Moreover, the proposition b ⩽ 2i + 7 is proven using Pick's Theorem with three equivalent inequalities. Furthermore, Scott's proof indicates that (x,y) = (3,3) is the only point where both upper bounds reach zero and the only case where equality can hold in Onion-Skin Theorem.