Tensor Products

The theory of supercharacters, recently developed by Diaconis-Isaacs and André, is used to derive the fundamental algebraic properties of Ramanujan sums. This machinery frequently yields one-line proofs of difficult identities and provides many novel formulas. In addition to exhibiting a new application of supercharacter theory, this article also serves as a blueprint for future work since some of the abstract results we develop are applicable in much greater generality.

In this paper, we study the vanishing-off subgroups of supercharacters, and use these to determine several new characterizations of supercharacter theory products. In particular, we give a character theoretic characterization that allows us to conclude that one may determine if a supercharacter theory is a ∆-product or * -product from the values of its corresponding supercharacters.

We relate the graph isomorphism problem to the solvability of certain systems of linear equations and linear inequalities. The number of these equations and inequalities is related to the complexity of the graphs isomorphism and subgraph isomorphim problems.

In this paper we propose two schemes of using the so-called QTTapproximation for the solution of multidimensional parabolic problems. First, we present a simple one-step implicit time integration scheme using an ALS-type solver in the QTT-format. As the second approach, we use the global space-time formulation, resulting in a large block linear system, encapsulating all time steps, and solve it at once in the QTT format. We prove the QTT-rank estimate for certain classes of multivariate potentials and respective solutions in (x, t) variables. The log-linear complexity of storage and the solution time is observed in both spatial and time grid sizes. The method is applied to the Fokker-Planck equation arising from the beadssprings models of polymeric liquids.

We prove that the lexicographic, degree lexicographic and the degree reverse lexicographic orders for monomials in R n = K[X 1 ,. .. , X n ] are uniquely determined by their induced orderings, (i.e. their restrictions to R n,i = K[X 1 ,. .. ,X i ,. .. , X n ]), when n 4. We also show that for any n 4 there are monomial orders that are not uniquely determined by their induced orderings, and provide examples of these orders for each n. the ustars conference proceedings 1 (2014), #R00

This thesis has been submitted in fulfilment of the requirements for a postgraduate degree (e.g. PhD, MPhil, DClinPsychol) at the University of Edinburgh. Please note the following terms and conditions of use: • This work is protected by copyright and other intellectual property rights, which are retained by the thesis author, unless otherwise stated. • A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. • This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the author. • The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the author. • When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given. A COMPUTATIONAL CLASSIFICATION OF MULTIVARIATE POLYNOMIALS USING SYMMETRIES AND REDUCTIONS.

In this paper we present a new method for solving the fractional Burgers equation u^{α}_{ t} − uu^{α}_{ x} = u^{αα}_{ x} , where u is a continuously differentiable function on [0, ∞). In this method an exact solution is obtained to the above equation using tensor product techniques. Exact solutions of nonlinear partial differential equations similar to this one are hard to find.

We introduce here a new method for extracting worst-cases of algorithms by using rewrite systems over automorphisms groups of inputs. We propose a canonical description of an algorithm, that is also related to the problem it solves. The description identifies an algorithm with a set of a rewrite systems over the automorphisms groups of inputs. All possible execution of the algorithm will then be reduced words of these rewriting system. Our main example is reducing two-dimensional Euclidean lattice bases. We deal with the Gaussian algorithm that finds shortest vectors in a two-dimensional lattice. We introduce four rewrite systems in the group of unimodular matrices, i.e. matrices with integer entries and with determinant equal to ±1 and deduce a new worst-case analysis of the algorithm that generalizes Vallée's result to the case of the usual Gaussian algorithm. An interesting (but not easy) future application will be lattice reduction in higher dimensions, in order to exhibit a tight upper-bound for the number of iterations of LLL-like reduction algorithms in the worst case. Sorting ordered finite sets are here as a nice esay example to illustrate the purpose of our method. We propose several rewrite systems in the group S of permutations and canonically identify a sorting algorithm with a rewrite system over S. This brings us to exhibit worst-cases of several sorting algorithms.

The Virasoro Lie algebra is a one-dimensional central extension of the Witt algebra, which can be realized as the Lie algebra of derivations on the algebra C[t ± ] of Laurent polynomials. Using this fact, we define a natural family of subalgebras of the Virasoro algebra, which we call polynomial subalgebras. We describe the one-dimensional modules for polynomial subalgebras, and we use this description to study the corresponding induced modules for the Virasoro algebra. We show that these induced modules are frequently simple and generalize a family of recently discovered simple modules. Additionally, we explore tensor products involving these induced modules. This allows us to describe new simple modules and also recover results on recently discovered simple modules for the Virasoro algebra.

We exhibit classes of polynomials whose sets of first partial derivatives form Gröbner bases, with respect to all term orders. The classes are defined by syntactic constraints on arithmetical formulas defining the polynomials. Read-once formulas without constants also have this property, while those with constants have a weaker "Gröbner-bounding" property introduced here. The same properties hold even with arbitrary powering of subterms of the formulas.

One of the greatest intellectual crimes to beset us in the 20th-century was the premature death of the formalist program. The millennia old dream of solving math was never realized as our efforts fell short of our ambition. David Hilbert, with all his grace, his effulgent brilliance, his professional magnanimity, and his unflinching dedication, was left with only disgruntled disappointment. The formalist program was a noble effort, held aloft by the unyielding conviction and charisma of the foremost mathematicians of the age. This forlorn vignette is rendered at least somewhat emotionally digestible by the developments that followed. The dream of syntax is all became partially realized by the contributions of Haskell Curry, Alonzo Church, Stephen Kleene, Moses Schönfinkel and others. With their syntactic approach to mathematics, we capture the compositional beauty of infinity in our humble symbol. And thus, we compile the syntactic face of God.

we present a new method for computing the real fixed points of polynomials using the resultants method. It is based on the theory of multi-resultants. The unstable calculation of the determinant of the large sparse matrix is replaced by a stable minimization problem using the Lanczos method. Thii technique will be able to take advantage of the sparseness of the resultant matrix. Algorithms and numerical results are presented.

Let 1 < g1 <. .. < g ϕ(p−1) < p − 1 be the ordered primitive roots modulo p. We study the pseudorandomness of the binary sequence (sn) defined by sn ≡ gn+1 + gn+2 mod 2, n = 0, 1,. . .. In particular, we study the balance, linear complexity and 2-adic complexity of (sn). We show that for a typical p the sequence (sn) is quite unbalanced. However, there are still infinitely many p such that (sn) is very balanced. We also prove similar results for the distribution of longer patterns. Moreover, we give general lower bounds on the linear complexity and 2-adic complexity of (sn) and state sufficient conditions for attaining their maximums. Hence, for carefully chosen p, these sequences are attractive candidates for cryptographic applications.

Let F q be the finite field of q elements, where q = p r is a power of the prime p, and (β 1 , β 2 ,. .. , β r) be an ordered basis of F q over F p. For ξ = r i=1 x i β i , x i ∈ F p , we define the Thue-Morse or sum-of-digits function T (ξ) on F q by T (ξ) = r i=1 x i. For a given pattern length s with 1 ≤ s ≤ q, a vector α = (α 1 ,. .. , α s) ∈ F s q with different coordinates α j 1 = α j 2 , 1 ≤ j 1 < j 2 ≤ s, a polynomial f (X) ∈ F q [X ] of degree d and a vector c = (c 1 ,. .. , c s) ∈ F s p we put T (c, α, f) = {ξ ∈ F q : T (f (ξ + α i)) = c i , i = 1,. .. , s}. In this paper we will see that under some natural conditions, the size of T (c, α, f) is asymptotically the same for all c and α in both cases, p → ∞ and r → ∞, respectively. More precisely, we have |T (c, α, f)| − p r−s ≤ (d − 1)q 1/2 under certain conditions on d, q and s. For monomials of large degree we improve this bound as well as we find conditions on d, q and s for which this bound is not true. In particular, if 1 ≤ d < p we have the dichotomy that the bound is valid if s ≤ d and for s ≥ d + 1 there are vectors c and α with T (c, α, f) = ∅ so that the bound fails for sufficiently large r. The case s = 1 was studied before by Dartyge and Sárközy.

In this note we recall (see also [8]) the structure of all recurrent sequences which satisfy a fixed recurrence relation, with entries in a perfect field. As a consequence of these considerations we give a reasonable proof for the known result that the Hadamard product of two recurrent sequences is also a recurrent sequence. Mathematics Subject Classification (2010): Primary 11B37,11B99; Secondary 47B37,47B99

Let K be an arbitrary algebraically closed field of characteristic zero and let K[[x]] be the ring of integral formal power series; let Ω be the K-subalgebra of K[[x]] generated by x and the subset TK = {exp(λx) : λ ∈ K}. In this note we supply some easy and elegant proofs for some classical results on the preimage of elements of the form xQ(x) exp(rx) through a linear differential operator with coefficients in K. We also make some theoretical considerations on the structure of the space of all solutions for a linear ODE defined over K[[x]]. 1. Some introductory remarks Let K be an algebraically closed field of characteristic zero [LS] and let x be a variable over K (simply an element x not belonging to K). LetK[[x]] be the ring of all formal integral power series f = a0+a1x+..., ai ∈ K, i = 0, 1, .... Here d dx : K[[x]] → K[[x]], df dx = a1 + 2a2x + ... + nanx n−1 + ..., is the usual differential operator defined on K[[x]]. For y ∈ K[[x]], we also denote y = dy dxn =   d dx ◦ .....

The paper contains a description of symmetric convolution of the algebra of block-symmetric analytic functions of bounded type on $\ell_{1}$-sum of the space $\mathbb{C}^{2}.$ We show that the specrum of such algebra does not coincide of point evaluation functionals and described characters of the algebra as functions of exponential type with plane zeros.

We describe a Guess-and-Check algorithm for computing algebraic equation invariants of the form ∧ifi(x1,. .. , xn) = 0, where each fi is a polynomial over the variables x1,. .. , xn of the program. The "guess" phase is data driven and derives a candidate invariant from data generated from concrete executions of the program. This candidate invariant is subsequently validated in a "check" phase by an off-the-shelf SMT solver. Iterating between the two phases leads to a sound algorithm. Moreover, we are able to prove a bound on the number of decision procedure queries which Guess-and-Check requires to obtain a sound invariant. We show how Guess-and-Check can be extended to generate arbitrary boolean combinations of linear equalities as invariants, which enables us to generate expressive invariants to be consumed by tools that cannot handle non-linear arithmetic. We have evaluated our technique on a number of benchmark programs from recent papers on invariant generation. Our results are encouraging-we are able to efficiently compute algebraic invariants in all cases, with only a few tests.

We develop a method that allows us to construct families of orthogonal matrix polynomials of size N × N satisfying second order difference equations with polynomial coefficients. The existence (and properties) of these orthogonal families strongly depends on the non commutativity of the matrix product, the existence of singular matrices and the matrix size N .

Teaching proofs of theorems and encouraging students to both comprehend written proofs and originate their own can at times be a difficult undertaking. This is due in part to the lack of a single unifying process by which one can approach mathematical proofs. In this paper a method using set theory as a foundation is presented.

We prove that the lexicographic, degree lexicographic and the degree reverse lexicographic orders for monomials in R n = K[X 1 ,. .. , X n ] are uniquely determined by their induced orderings, (i.e. their restrictions to R n,i = K[X 1 ,. .. ,X i ,. .. , X n ]), when n 4. We also show that for any n 4 there are monomial orders that are not uniquely determined by their induced orderings, and provide examples of these orders for each n. the ustars conference proceedings 1 (2014), #R00

We study the theory of projective reconstruction for multiple projections from an arbitrary dimensional projective space into lower-dimensional spaces. This problem is important due to its applications in the analysis of dynamical scenes. The current theory, due to Hartley and Schaffalitzky, is based on the Grassmann tensor, generalizing the ideas of fundamental matrix, trifocal tensor and quadrifocal tensor used in the wellstudied case of 3D to 2D projections. We present a theory whose point of departure is the projective equations rather than the Grassmann tensor. This is a better fit for the analysis of approaches such as bundle adjustment and projective factorization which seek to directly solve the projective equations. In a first step, we prove that there is a unique Grassmann tensor corresponding to each set of image points, a question that remained open in the work of Hartley and Schaffalitzky. Then, we prove that projective equivalence follows from the set of projective equations given certain conditions on the estimated camera-point setup or the estimated projective depths. Finally, we demonstrate how wrong solutions to the projective factorization problem can happen, and classify such degenerate solutions based on the zero patterns in the estimated depth matrix.

We study the theory of projective reconstruction for multiple projections from an arbitrary dimensional projective space into lower-dimensional spaces. This problem is important due to its applications in the analysis of dynamical scenes. The current theory, due to Hartley and Schaffalitzky, is based on the Grassmann tensor, generalizing the ideas of fundamental matrix, trifocal tensor and quadrifocal tensor used in the wellstudied case of 3D to 2D projections. We present a theory whose point of departure is the projective equations rather than the Grassmann tensor. This is a better fit for the analysis of approaches such as bundle adjustment and projective factorization which seek to directly solve the projective equations. In a first step, we prove that there is a unique Grassmann tensor corresponding to each set of image points, a question that remained open in the work of Hartley and Schaffalitzky. Then, we prove that projective equivalence follows from the set of projective equations given certain conditions on the estimated camera-point setup or the estimated projective depths. Finally, we demonstrate how wrong solutions to the projective factorization problem can happen, and classify such degenerate solutions based on the zero patterns in the estimated depth matrix.

This paper presents a generalized version of the classic projective reconstruction theorem which helps to choose or assess depth constraints for projective depth estimation algorithms. The theorem shows that projective reconstruction is possible under a much weaker restriction than requiring all estimated projective depths to be nonzero. This result enables us to present a class of depth constraints under which any reconstruction of cameras and points projecting into given image points is projectively equivalent to the true camera-point configuration. It also completely specifies the possible wrong configurations allowed by other constraints. We demonstrate the application of the theorem by analysing several constraints used in the literature, as well as presenting new constraints with desirable properties. We mention some of the implications of our results on iterative depth estimation algorithms and projective reconstruction via rank minimization. Our theory is verified by running experiments on both synthetic and real data. Keywords multiple view geometry • projective reconstruction • Projective Reconstruction Theorem • projective factorization • projective depths • depth constraints NICTA is funded by the Australian Government as represented by the Department of Broadband, Communications and the Digital Economy and the Australian Research Council through the ICT Centre of Excellence program.

In this article we consider the iterative schemes to compute the canonical (CP) approximation of quantized data generated by a function discretized on a large uniform grid in an interval on the real line. This paper continues the research on the QTT method [16] developed for the tensor train (TT) approximation of the quantized images of function related data. In the QTT approach the target vector of length 2 L is reshaped to a L th order tensor with two entries in each mode (Quantized representation) and then approximated by the QTT tenor including 2r 2 L parameters, where r is the maximal TT rank. In what follows, we consider the Alternating Least-Squares (ALS) iterative scheme to compute the rank-r CP approximation of the quantized vectors, which requires only 2rL ≪ 2 L parameters for storage. In the earlier papers [17] such a representation was called Q Can format, while in this paper we abbreviate it as the QCP representation. We test the ALS algorithm to calculate the QCP approximation on various functions, and in all cases we observed the exponential error decay in the QCP rank. The main idea for recovering a discretized function in the rank-r QCP format using the reduced number the functional samples, calculated only at O(2rL) grid points, is presented. The special version of ALS scheme for solving the arising minimization problem is described. This approach can be viewed as the sparse QCP-interpolation method that allows to recover all 2rL representation parameters of the rank-r QCP tensor. Numerical examples show the efficiency of the QCP-ALS type iteration and indicate the exponential convergence rate in r.

In 1985 Matter introduced the ideal Ppσ of pσ−absolutely continuous operators with pa-rameters 1 ≤ p &lt; ∞ and 0 ≤ σ &lt; 1 (see [2] and [3]). In 1997 Sanchéz [12] presented a new characterization of Ppσ by means of an associated tensor norm. In this paper we ...

A standard method for finding a rational number from its values modulo a collection of primes is to determine its value modulo the product of the primes via Chinese remaindering, and then use Farey sequences for rational reconstruction. Successively enlarging the set of primes if needed, this method is guaranteed to work if we restrict ourselves to "good" primes. Depending on the particular application, however, there may be no efficient way of identifying good primes. In the algebraic and geometric applications we have in mind, the final result consists of an a priori unknown ideal (or module) which is found via a construction yielding the (reduced) Gröbner basis of the ideal. In this context, we discuss a general setup for modular and, thus, potentially parallel algorithms which can handle "bad" primes. A key new ingredient is an error tolerant algorithm for rational reconstruction via Gaussian reduction.

We present a polynomial time randomized algorithm for reconstructing $\Sigma\Pi\Sigma(2)$ circuits over $\mathbb{R}$, i.e. depth 3 circuits with fan in 2 at the top addition gate and having real coefficients. The algorithm needs only a blackbox query access to the polynomial $f\in \mathbb{R}[x_1,\ldots,x_n]$ of degree d in n variables, computable by a $\Sigma\Pi\Sigma(2)$ circuit C. In addition, we assume that the simple rank of this polynomial (essential number of variables after removing the gcd of the two multiplication gates) is bigger than a fixed constant. Our algorithm runs in time $poly(n,d)$ and returns an equivalent $\Sigma\Pi\Sigma(2)$ circuit(with high probability). Our main techniques are based on the use of Quantitative Syslvester Gallai Theorems from the work of Barak et.al.([3]) to find a small collection of nice subspaces to project onto. The heart of our paper lies in subtle applications of the Quantitative Sylvester Gallai theorems to prove why projections w.r.t. ...

Reconstruction of arithmetic circuits has been heavily studied in the past few years and has connections to proving lower bounds and deterministic identity testing. In this paper we present a polynomial time randomized algorithm for reconstructing $\Sigma\Pi\Sigma(2)$ circuits over $\mathbb{F}$ ($char(\mathbb{F})=0$), i.e. depth$-3$ circuits with fan-in $2$ at the top addition gate and having coefficients from a field of characteristic $0$. The algorithm needs only a blackbox query access to the polynomial $f \in \mathbb{F}[x_1,\ldots, x_n]$ of degree $d$, computable by a $\Sigma\Pi\Sigma(2)$ circuit $C$. In addition, we assume that &quot;simple rank&quot; of this polynomial (essential number of variables after removing gcd of the two multiplication gates) is bigger than a constant. Our algorithm runs in time $poly(n, d)$ and returns an equivalent $\Sigma\Pi\Sigma(2)$ circuit(with high probability). The problem of reconstructing $\Sigma\Pi\Sigma(2)$ circuits over finite fields was f...

In this paper we develop efficient randomized algorithms to solve the black-box reconstruction problem for polynomials over finite fields, computable by depth three arithmetic circuits with alternating addition/multiplication gates, such that output gate is an addition gate with in-degree two. Such circuits naturally compute polynomials of the form G× (T1 + T2), where G, T1, T2 are product of affine forms computed at the first layer in the circuit, and polynomials T1, T2 have no common factors. Rank of such a circuit is defined to be the dimension of vector space spanned by all affine factors of T1 and T2. For any polynomial f computable by such a circuit, rank( f ) is defined to be the minimum rank of any such circuit computing it. Our work develops randomized reconstruction algorithms which take as input black-box access to a polynomial f (over finite field F), computable by such a circuit. Here are the results. • [Low rank] : When 5 ≤ rank( f ) = O(log d), it runs in time (ndlog ...

This manuscript presents a generalization of the structure of the null space of the Bezout matrix in the monomial basis, see [15], to an arbitrary basis. In addition, two methods for computing the gcd of several polynomials, using also Bezout matrices, without having to convert them to the monomial basis. The main point is that the presented results are expressed with respect to an arbitrary polynomial basis. In recent years, many problems in polynomial systems, stability theory, CAGD, etc., are solved using Bezout matrices in distinct specific bases. Therefore, it is very useful to have results and tools that can be applied to any basis.

This paper is devoted to introducing a new approach for computing the topology of a real algebraic plane curve presented either parametrically or defined by its implicit equation when the corresponding polynomials which describe the curve are known only "by values". This approach is based on the replacement of the usual algebraic manipulation of the polynomials (and their roots) appearing in the topology determination of the given curve with the computation of numerical matrices (and their eigenvalues). Such numerical matrices arise from a typical construction in Elimination Theory known as the Bézout matrix which in our case is specified by the values of the defining polynomial equations on several sample points.

The availability of the implicit equation of a plane curve or of a 3D surface can be very useful in order to solve many geometric problems involving the considered curve or surface: for example, when dealing with the point position problem or answering intersection questions. On the other hand, it is well known that in most cases, even for moderate degrees, the implicit equation is either difficult to compute or, if computed, the high degree and the big size of the coefficients makes extremely difficult its use in practice. We will show that, for several problems involving plane curves, 3D surfaces and some of their constructions (for example, offsets), it is possible to use the implicit equation (or, more precisely, its properties) without needing to explicitly determine it. We replace the computation of the implicit equation by the evaluation of the considered parameterizations in a set of points and its use, in order to deal with the considered geometric problems, is translated into one or several generalized eigenvalue problems on matrix pencils (depending again on several evaluations of the considered parameterizations). This is the so called "Polynomial Algebra by Values" approach where the huge polynomial equations coming from Elimination Theory (e.g., using resultants) are replaced by big structured and sparse numerical matrices. For these matrices there are well known numerical techniques allowing to provide the results we need to answer the geometric questions on the considered curves and surfaces.

In this paper we describe some properties of companion matrices and demonstrate some special patterns that arisewhen a Toeplitz or a Hankel matrix is multiplied by a related companion matrix.We present a necessary and sufficient condition, generalizing known results, for a matrix to be the transforming matrix for a similarity between a pair of companion matrices. A special case of our main result shows that a Toeplitz or a Hankel matrix can be extended using associated companion matrices, preserving the Toeplitz or Hankel structure respectively.

where , the existential quantifier, denotes “there exists’’ and is the conjunction that denotes “and.’’ Notice that while both sides of the equivalence, , are in fact equivalent, the right hand side does not involve y, i.e., the variable y has been eliminated. In fact, the quantifier has been also eliminated; for this reason, the procedure is also called “quantifier elimination.’’ A serious deficiency of Gaussian elimination is that it works only for an existentially quantified conjunction of linear equations. However, many real-life scientific and engineering problems deal with nonlinear equations, inequalities ( , &gt;, . . .), disjunctions ( , which stands for “or’’), implications ( ), and the universal quantifier (∀ , which denotes “for all”). In studying the stability of a numerical method for solving a certain PDE, we encounter the following expression:

We study a generalization of the classical correspondence between homogeneous quadratic polynomials, quadratic forms, and symmetric/alternating bilinear forms to forms in n variables. The main tool is combinatorial polarization, and the approach is applicable even when n! is not invertible in the underlying field.

Let F(x,u1,...,u&#x27;) be a square-free polynomial which is monic w.r.t. x and let (s1,...,s&#x27;) 2 C &#x27; . If F(x,s1,...,s&#x27;) is square-free then the roots of F w.r.t. x can be expanded into integral power series in u1 ¡ s1,...,u&#x27; ¡ s&#x27;, and the power series roots can be calculated by well-known Newton&#x27;s method. We use the floating-point number arithmetic to calculate the numerical coecients, and this paper investigates the numerical errors contained in the roots computed. We first express the power series root in terms of sub-polynomials of F(x,s1+v1,...,s&#x27;+v&#x27;), with (v1,...,v&#x27;) a set of new variables, and clarify various properties of the power series roots. Then, we investigate the cancellation errors when the expansion point (s1,...,s&#x27;) is close to a &quot;singular point&quot; and far from the origin, respectively. We find that, although the cancellation errors are not large usually, they become extremely large in some cases. In fact,...

Let $\mathbb{F}$ be an infinite field. Let $n$ be a positive integer and let $1\leq d\leq n$. Let $\vec{f}_1, \vec{f}_2, \ldots, \vec{f}_{d-1} \in \mathbb{F}^{n}$ be $d-1$ linearly independent vectors. Let $\vec{x}=(x_1,x_2,\ldots,x_{d},0,0,\ldots,0)\in\mathbb{F}^{n}$, with $n-d$ zeros at the end. Let $\vec{R}: \mathbb{F}^n \to\mathbb{F}^n$ be the cyclic shift operator to the right, e.g. $\vec{R}\,\vec{x} = (0,x_1,x_2,\ldots,x_{d},0,0,\ldots,0)$. Is there a vector $\vec{x} \in \mathbb{F}^n$, such that the $n-d+1$ vectors $\vec{x},\vec{R}\vec{x}, \ldots ,\vec{R}^{n-d}\vec{x}$ complete the set $\{\vec{f}_j\}_{j=1}^{d-1}$ to a basis of $\mathbb{F}^n$? The answer is in the affirmative for every linearly independent set of $\vec{f}_j$, $j=1,2,\ldots,d-1$. In order to prove this fact, we prove that the $(n-d+1)\times(n-d+1)$ minors of the $(n-d+1)\times(n-d+1)$ circulant matrix. $\begin{bmatrix} \vec{x}, \vec{R} \vec{x}, \ldots, \vec{R}^{n-d} \vec{x} \end{bmatrix}^\intercal$ form a Grobne...

Let P(t) ∈ K(t) n be a rational parametrization of an algebraic space curve C. In this paper, we introduce the notion of limit point, P L , of the given parametrization P(t), and some remarkable properties of P L are obtained. In addition, we generalize the results in [2] concerning the T-function, T (s), which is defined by means of a univariate resultant. More precisely, independently on whether the limit point is regular or not, we show that T (s) = n i=1 H P i (s) m i −1 , where the polynomials H P i (s), i = 1,. .. , n are the fibre functions of the singularities P i ∈ C of multiplicity m i , i = 1,. .. , n. The roots of H P i (s) are the fibre of P i for i = 1,. .. , n. Thus, a complete classification of the singularities of a given space curve, via the factorization of a resultant, is obtained.

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Suppose f is a map from a non-empty nite set X to a nite group G. Dene the map f G : G ! N ( f0g by g 7! jf 1(g)j. In this article, we show that for a suitable choice of f, the map f G is a character. We use our results to show that the solution function for the word equation w(t1;t2;:::;tn) = g (g 2 G) is a character, where w(t1;t2;:::;tn) denotes the product of t1;t2;:::;tn;t 1 1 ;t 1

Let T be an algebraic torus, defined over any field k, and let V be a vector space over k. Let ρ be a finite dimensional regular linear representation of T on V . Our goal is to show that there exists a number d0 such that, for all n ≥ 1, the invariant ring: Rn = [⊕d≥0 ⊗ Sym(V )] is generated by elements of degree ≤ d0. Before we begin, let’s make a few notes. If we let r be the dimension of T , then T = Spec{k[t1, t−1 1 , . . . , tr, t−1 r ]}. Next, if χ(t1, . . . , tr) = t a1 1 . . . t ar r , where each ai ∈ Z, then let Uχ to be the one-dimensional representation of T such that: (t1, . . . , tr) · u = χ(t1, . . . , tr) · u where u ∈ Uχ. We begin by choosing a convenient basis for our vector space V . To do this, we appeal to the following theorem:

We define a switch function to be a function from an interval to {1, −1} with a finite number of sign changes. (Special cases are the Walsh functions.) By a topological argument, we prove that, given n real-valued functions, f 1 ,. .. , f n , in L 1 [0, 1], there exists a switch function, σ, with at most n sign changes that is simultaneously orthogonal to all of them in the sense that 1 0 σ(t)f i (t)dt = 0, for all i = 1,. .. , n. Moreover, we prove that, for each λ ∈ (−1, 1), there exists a unique switch function, σ, with n switches such that 1 0 σ(t)p(t)dt = λ 1 0 p(t)dt for every real polynomial p of degree at most n − 1. We also prove the same statement holds for every real even polynomial of degree at most 2n − 2. Furthermore, for each of these latter results, we write down, in terms of λ and n, a degree n polynomial whose roots are the switch points of σ; we are thereby able to compute these switch functions.