small fixes on inputs & outputs

A1-Brouwer/Documentation/AnisotropicDimensionDoc.m2

@@ -1,13 +1,13 @@
 document{
     Key => {anisotropicDimensionQp, (anisotropicDimensionQp, GrothendieckWittClass, ZZ)},
-    Headline => "Returns the anisotropic dimension of a rational symmetric bilinear form over the p-adics",
+    Headline => "returns the anisotropic dimension of a rational symmetric bilinear form over the p-adics",
     Usage => "anisotropicDimensionQp(beta, p)",
     Inputs => {
-	GrothendieckWittClass => "beta" => {"Any class ", TEX///$\beta\in\text{GW}(\mathbb{Q})$///, ". "},
-	ZZ => "p" => {"A prime number"},
+	GrothendieckWittClass => "beta" => {"over ", TEX///$\mathbb{Q}$///},
+	ZZ => "p" => {"a prime number"},
 	},
     Outputs => {
-        ZZ => {"The rank of the anisotropic part of ", TEX///$\beta$///, " over ", TEX///$\mathbb{Q}_p$///, "."},
+        ZZ => {"the rank of the anisotropic part of ", TEX///$\beta$///, " over ", TEX///$\mathbb{Q}_p$///},
 	},
     PARA{"This is an implementation of [KC18, Algorithm 8] in Macaulay2, which computes the anisotropic dimension of rational forms over the ", TEX///$p$///,"-adics. Note that any form of rank ", TEX///$\ge 5$///, " is always isotropic, so this method will return 0, 1, 2, 3, or 4."},
     PARA{EM "Citations:"},
@@ -60,10 +60,10 @@ document{
     Headline => "Returns the Witt index of a symmetric bilinear form",
     Usage => "WittIndex(beta)",
     Inputs => {
-	GrothendieckWittClass => "beta" => {"Any class ", TEX///$\beta\in\text{GW}(k)$///, " where ", TEX///$k$///, " is the complex numbers, reals, rationals, or a finite field."},
+	GrothendieckWittClass => "beta" => {"denoted by ", TEX///$\beta\in\text{GW}(k)$///, ", where ", TEX///$k$///, " is the complex numbers, reals, rationals, or a finite field"},
 	},
     Outputs => {
-        ZZ => {"The rank of the totally isotropic part of ", TEX///$\beta$///, "."},
+        ZZ => {"the rank of the totally isotropic part of ", TEX///$\beta$///},
 	},
     PARA{"By Witt decomposition, any form decomposes uniquely as ", TEX///$\beta \cong k \mathbb{H} \oplus \beta_a$///," where the form ", TEX///$\beta_a$///," is anisotropic. The integer ", TEX///$k$///, " is called the ", EM "Witt index", " of ", TEX///$\beta$///, ". See for instance [L05, I.4.3]."},
 

A1-Brouwer/Documentation/ArithmeticMethodsDoc.m2

@@ -43,11 +43,11 @@ document{
     Headline => "p-adic valuation of a rational number or integer",
     Usage => "padicValuation(a, p)",
     Inputs => {
-	ZZ => "a" => {"A non-zero rational number in ", TEX///$\mathbb{Q}_p$///, "."},
-	ZZ => "p" => {"A rational prime number."},
+	ZZ => "a" => {"a non-zero rational number in ", TEX///$\mathbb{Q}_p$///},
+	ZZ => "p" => {"a rational prime number"},
 	},
     Outputs =>{
-	ZZ => {"An integer ", TEX///$n$///, " where ",TEX///$a=p^n u$///, " and ", TEX///$u$///," is a unit in ", TEX///$\mathbb{Z}_p$///},
+	ZZ => {"an integer ", TEX///$n$///, " where ",TEX///$a=p^n u$///, " and ", TEX///$u$///," is a unit in ", TEX///$\mathbb{Z}_p$///},
         },
     EXAMPLE lines///
     a = 363/7;
@@ -61,11 +61,11 @@ document {
 	Headline => "produces a basis for a local finitely generated algebra over a field k",
 	Usage => "localAlgebraBasis(L,p)",
 	Inputs => {
-	    List => "L" => {"list of polynomials ", TEX///$f=(f_1, \dots ,f_n)$///, " over the same ring"},
-	    Ideal => "p" => {"prime ideal of an isolated zero"}
+	    List => "L" => {"of polynomials ", TEX///$f=(f_1, \dots ,f_n)$///, " over the same ring"},
+	    Ideal => "p" => {"a prime ideal of an isolated zero"}
 	    },
 	Outputs => {
-	    List => {"a list of basis elements of the local k-algebra ", TEX///$Q_p(f)$/// }
+	    List => {"of basis elements of the local ",TEX///$k$///,"-algebra ", TEX///$Q_p(f)$/// }
 	    },
 	PARA {"Given an endomorphism of affine space, ", TEX///$f=(f_1,\dots ,f_n)$///,
 			", given as a list of polynomials called ", TT "L", " and the prime ideal of an isolated zero, this command returns a list of basis elements of the local k-algebra ", TEX///$Q_p(f)$///, " by computing a normal basis for ", TEX///$(I:(I:p^{\infty}))$///, " (vis. [S02, Proposition 2.5])."},

A1-Brouwer/Documentation/BuildingFormsDoc.m2

@@ -6,7 +6,7 @@ document {
 	Inputs => {
 	    Ring => "k" => {"a field"},
 	    RingElement => "a" => {"any element ", TEX///$a\in k$///},
-	    Sequence => "L" => {"a list of elements ", TEX///$L = (a_1,\ldots,a_n)$///, " with ", TEX///$a_i \in k$///},
+	    Sequence => "L" => {"of elements ",TEX///$a_{i} \in k$///, ", where ", TEX///$i = 1,\dots, n$///},
 	    }, 
 	Outputs => { 
 	    GrothendieckWittClass => {"the diagonal form ", TEX///$\langle a_1,\ldots,a_n\rangle \in \text{GW}(k)$///},
@@ -31,7 +31,7 @@ document {
 	Inputs => {
 	    Ring => "k" => {"a field"},
 	    RingElement => "a" => {"any element ", TEX///$a\in k$///},
-	    Sequence => "L" => {"a list of elements ", TEX///$L = (a_1,\ldots,a_n)$///, " with ", TEX///$a_i \in k$///},
+	    Sequence => "L" => {"of elements ", TEX///$L = (a_1,\ldots,a_n)$///, " with ", TEX///$a_i \in k$///},
 	    }, 
 	Outputs => { 
 	    GrothendieckWittClass => {"the Pfister form ", TEX///$\langle\langle a_1,\ldots,a_n\rangle\rangle \in \text{GW}(k)$///},

A1-Brouwer/Documentation/DecompositionDoc.m2

@@ -3,7 +3,7 @@ document {
     Headline => "produces a simplified diagonal representative of a Grothendieck Witt class",
     Usage => "sumDecomposition(beta)",
     Inputs => {
-        GrothendieckWittClass => "beta" => {"a symmetric bilinear form defined over a field ", TEX///$k$///, "."},
+        GrothendieckWittClass => "beta" => {"a symmetric bilinear form defined over a field ", TEX///$k$///},
     },
     Outputs => { 
 	GrothendieckWittClass => {"a diagonal representative of the Grothendieck Witt class of the input form"},
@@ -34,11 +34,11 @@ document {
     Headline => "produces a simplified diagonal representative of a Grothendieck Witt class",
     Usage => "sumDecompositionString(beta)",
     Inputs => {
-        GrothendieckWittClass => "beta" => {"a symmetric bilinear form defined over a field ", TEX///$k$///, "."},
+        GrothendieckWittClass => "beta" => {"a symmetric bilinear form defined over a field ", TEX///$k$///},
     },
     Outputs => { 
 	--GrothendieckWittClass => {"a diagonal representative of the Grothendieck Witt class of the input form"},
-	String => {"The decomposition as a sum of hyperbolic and rank one forms."},
+	String => {"the decomposition as a sum of hyperbolic and rank one forms"},
 	},
     PARA {"Given a symmetric bilinear form ", TT"beta", " over a field ", TEX///$k$///, ", we return a simplified diagonal form of ", TT"beta","."},
     EXAMPLE lines ///
@@ -67,7 +67,7 @@ document {
     Headline => "returns the anisotropic part of a Grothendieck Witt class",
     Usage => "anisotropicPart(beta)",
     Inputs => {
-        GrothendieckWittClass => "beta" => {"a symmetric bilinear form defined over a field ", TEX///$k$///, "."},
+        GrothendieckWittClass => "beta" => {"a symmetric bilinear form defined over a field ", TEX///$k$///},
     },
     Outputs => { 
 	GrothendieckWittClass => {"the anisotropic part of ", TEX///$\beta$///},

A1-Brouwer/Documentation/GWInvariantsDoc.m2

@@ -3,10 +3,10 @@ document{
     Headline => "Outputs the signature of a symmetric bilinear form over the real or rational numbers",
     Usage => "signature(beta)",
     Inputs => {
-	GrothendieckWittClass => "beta" => {"A symmetric bilinear form defined over ", TEX///$\mathbb{Q}$///, " or ", TEX///$\mathbb{R}$///, "."},
+	GrothendieckWittClass => "beta" => {"a symmetric bilinear form defined over ", TEX///$\mathbb{Q}$///, " or ", TEX///$\mathbb{R}$///},
 	},
     Outputs => {
-	ZZ => "n" => {"The ", EM "signature", " of the symmetric bilinear form ", TEX///$\beta$///, "."},
+	ZZ => "n" => {"the ", EM "signature", " of the symmetric bilinear form ", TEX///$\beta$///},
 	},
     PARA{"Given a symmetric bilinear form, after diagonalizing it, we can consider the number of positive entries minus the number of negative entries appearing along the diagonal. This is the ", EM "signature", " of a symmetric bilinear form, and is one of the primary invariants we use to classify forms. For more information see ", TO2(gwIsomorphic,"gwIsomorphic"), "."},
     EXAMPLE lines ///
@@ -22,10 +22,10 @@ document{
     Headline => "outputs an integral discriminant for a rational symmetric bilinear form",
     Usage => "integralDiscriminant(beta)",
     Inputs => {
-	GrothendieckWittClass => "beta" => {"Any class ", TEX///$\beta\in\text{GW}(\mathbb{Q})$///, "."},
+	GrothendieckWittClass => "beta" => {"denoted by ", TEX///$\beta\in\text{GW}(\mathbb{Q})$///},
 	},
     Outputs => {
-        ZZ => {"An integral square class representative of ", TEX///$\text{disc}(\beta)$///, "."},
+        ZZ => {"an integral square class representative of ", TEX///$\text{disc}(\beta)$///},
 	},
     EXAMPLE lines ///
     beta = gwClass(matrix(QQ,{{1,4,7},{4,3,-1},{7,-1,5}}));
@@ -39,11 +39,11 @@ document{
     Headline => "outputs the Hasse-Witt invariant for a prime p for the quadratic form of the Grothendieck-Witt class",
     Usage => "HasseWittInvariant(beta, p)",
     Inputs => {
-	GrothendieckWittClass => "beta" => {"Any class ", TEX///$\beta\in\text{GW}(\mathbb{Q})$///, "."},
-	ZZ => "p" => {"A integral prime number."},
+	GrothendieckWittClass => "beta" => {"denoted by ", TEX///$\beta\in\text{GW}(\mathbb{Q})$///},
+	ZZ => "p" => {"an integral prime number"},
 	},
     Outputs => {
-        ZZ => {"The Hasse-Witt invariant for ", TEX///$\beta$///," for the prime ",TEX///$p$///},
+        ZZ => {"the Hasse-Witt invariant for ", TEX///$\beta$///," for the prime ",TEX///$p$///},
 	},
     PARA{"The ", EM "Hasse-Witt invariant", " of a diagonal form ", TEX///$\langle a_1,\ldots,a_n\rangle$///, " over a field ", TEX///$K$///, " is defined to be the product ", TEX///$\prod_{i<j} \left((a_i,a_j)_p \right)$///, " where ", TEX///$(-,-)_p$///, " is the ", TO2(HilbertSymbol,"Hilbert symbol"), "."},
     PARA{"The Hasse-Witt invariant of a form will be equal to 1 for almost all primes. In particular after diagonalizing a form ", TEX///$\beta \cong \left\langle a_1,\ldots,a_n\right\rangle$///, " then the Hasse-Witt invariant at a prime ", TEX///$p$///, " will be 1 automatically if ", TEX///$p\nmid a_i$///, " for all ", TEX///$i$///, ". Thus we only have to compute the invariant at ", TO2(relevantPrimes, "primes dividing diagonal entries"),  "."},
@@ -60,10 +60,10 @@ document{
     Headline => "outputs a list of primes at which the Hasse-Witt invariants of a symmetric bilinear form may be non-trivial",
     Usage => "relevantPrimes(beta)",
     Inputs => {
-	GrothendieckWittClass => "beta" => {"Any class ", TEX///$\beta\in\text{GW}(\mathbb{Q})$///, "."},
+	GrothendieckWittClass => "beta" => {"denoted by ", TEX///$\beta\in\text{GW}(\mathbb{Q})$///},
 	},
     Outputs => {
-        List => {"A finite list of primes ", TEX///$(p_1,\ldots,p_r)$///, " for which the Hasse-Witt invariants ", TEX///$\phi_p(\beta)$///," may be nontrivial."},
+        List => {"a finite list of primes ", TEX///$(p_1,\ldots,p_r)$///, " for which the Hasse-Witt invariants ", TEX///$\phi_p(\beta)$///," may be nontrivial"},
 	},
     PARA{"It is a classical result that the ", TO2(HasseWittInvariant,"Hasse-Witt invariants"), " of a quadratic form are equal to 1 for all but finitely many primes (see e.g. [S73, IV Section 3.3]. As the Hasse-Witt invariants are computed as a product of ", TO2(HilbertSymbol,"Hilbert symbols") , " of the pairwise entries appearing on a diagonalization of the symbol, it suffices to consider primes dividing diagonal entries."},
    EXAMPLE lines ///

A1-Brouwer/Documentation/GrothendieckWittClassesDoc.m2

@@ -34,7 +34,7 @@ document {
 	    Matrix => "M" => {"a symmetric matrix defined over an arbitrary field"}
 	    },
 	Outputs => {
-	    GrothendieckWittClass => { "the isomorphism class of a symmetric bilinear form represented by ", TEX/// $M$///, "." }
+	    GrothendieckWittClass => { "the isomorphism class of a symmetric bilinear form represented by ", TEX/// $M$///}
 	    },
 	PARA {"Given a symmetric matrix, ", TEX///$M$///, ", this command outputs an object of type ", TT "GrothendieckWittClass", ". ",
                 "This output has the representing matrix, ", TEX///$M$///, ", and the base field of the matrix stored in its CacheTable."},

A1-Brouwer/Documentation/HilbertSymbolsDoc.m2

@@ -3,12 +3,12 @@ document{
     Headline => "Computes the Hilbert symbol of two integers or rational numbers at a prime",
     Usage => "HilbertSymbol(a,b,p)",
     Inputs => {
-	QQ => "a" => {"Any integer or rational number, considered as an element of ", TEX///$\mathbb{Q}_p$///, "."},
-	QQ => "b" => {"Any integer or rational number, considered as an element of ", TEX///$\mathbb{Q}_p$///, "."},
-	ZZ => "p" => {"Any integer prime number."},
+	QQ => "a" => {"any integer or rational number, considered as an element of ", TEX///$\mathbb{Q}_p$///},
+	QQ => "b" => {"any integer or rational number, considered as an element of ", TEX///$\mathbb{Q}_p$///},
+	ZZ => "p" => {"any integer prime number"},
 	},
     Outputs => {
-	ZZ => {"The ", EM "Hilbert symbol ", TEX///$(a,b)_p$///, "."},
+	ZZ => {"the ", EM "Hilbert symbol ", TEX///$(a,b)_p$///},
 	},
     PARA{"The ", EM "Hasse-Witt invariant", " of a diagonal form ", TEX///$\langle a_1,\ldots,a_n\rangle$///, " over a field ", TEX///$K$///, " is defined to be the product ", TEX///$\prod_{i<j}  \phi(a_i,a_j)$///, " where ", TEX///$\phi \colon K \times K \to \left\{\pm 1\right\}$///, " is any ", EM "symbol", " (see e.g. [MH73, III.5.4] for a definition). It is a classical result of Hilbert that over a local field of characteristic not equal to two, there is one and only symbol, ", TEX///$(-,-)_p$///,  " called the ", EM "Hilbert symbol", " ([S73, Chapter III]) computed as follows:"},
     PARA{TEX///$(a,b)_p = \begin{cases} 1 & z^2 = ax^2 + by^2 \text{ has a nonzero solution in } K^3 \\ -1 & \text{otherwise.} \end{cases}$///},
@@ -30,11 +30,11 @@ document{
     Headline => "Computes the Hilbert symbol of two rational numbers over the reals",
     Usage => "HilbertSymbolReal(a,b,p)",
     Inputs => {
-	QQ => "a" => {"Any non-zero integer or rational number, considered as an element of ", TEX///$\mathbb{Q}_p$///, "."},
-	QQ => "b" => {"Any non-zero integer or rational number, considered as an element of ", TEX///$\mathbb{Q}_p$///, "."},
+	QQ => "a" => {"any non-zero integer or rational number, considered as an element of ", TEX///$\mathbb{Q}_p$///},
+	QQ => "b" => {"any non-zero integer or rational number, considered as an element of ", TEX///$\mathbb{Q}_p$///},
 	},
     Outputs => {
-	ZZ => {"The ", EM "Hilbert symbol ", TEX///$(a,b)_{\mathbb{R}}$///, "."},
+	ZZ => {"the ", EM "Hilbert symbol ", TEX///$(a,b)_{\mathbb{R}}$///},
 	},
     PARA{"The ", EM "Hasse-Witt invariant", " of a diagonal form ", TEX///$\langle a_1,\ldots,a_n\rangle$///, " over a field ", TEX///$K$///, " is defined to be the product ", TEX///$\prod_{i<j}  \phi(a_i,a_j)$///, " where ", TEX///$\phi \colon K \times K \to \left\{\pm 1\right\}$///, " is any ", EM "symbol", " (see e.g. [MH73, III.5.4] for a definition). It is a classical result of Hilbert that over a local field of characteristic not equal to two, there is one and only symbol, ", TEX///$(-,-)_p$///,  " called the ", EM "Hilbert symbol", " ([S73, Chapter III]) computed as follows:"},
     PARA{TEX///$(a,b)_{\mathbb{R}} = \begin{cases} 1 & z^2 = ax^2 + by^2 \text{ has a nonzero solution in } {\mathbb{R}}^3 \\ -1 & \text{otherwise.} \end{cases}$///},

A1-Brouwer/Documentation/IsomorphismOfFormsDoc.m2

@@ -3,8 +3,8 @@ document{
     Headline => "Determines whether two Grothendieck Witt classes are isomorphic over CC, RR, QQ, or a finite field.",
     Usage => "gwIsomorphic(alpha,beta)",
     Inputs => {
-	GrothendieckWittClass => "alpha" => {"Any Grothendieck-Witt class ", TEX///$\alpha$///, "."},
-	GrothendieckWittClass => "beta" => {"Any Grothendieck-Witt class ", TEX///$\beta$///, "."},
+	GrothendieckWittClass => "alpha" => {"denoted by ",TEX///$\alpha$///},
+	GrothendieckWittClass => "beta" => {"denoted by ",TEX///$\beta$///},
 	},
     Outputs => {
 	Boolean => {"returns true or false depending on whether two Grothendieck Witt classes are equal in the Grothendieck-Witt ring"},

A1-Brouwer/Documentation/IsotropyDoc.m2

@@ -3,10 +3,10 @@ document{
     Headline => "Determines whether a Grothendieck-Witt class is isotropic",
     Usage => "isIsotropic(beta)",
     Inputs => {
-	GrothendieckWittClass => "beta" => {"Any class ", TEX///$\beta\in\text{GW}(k)$///, " where ", TEX///$k$///, " is the rationals, reals, complex numbers, or a finite field."},
+	GrothendieckWittClass => "beta" => {"denoted by ", TEX///$\beta\in\text{GW}(k)$///, ", where ", TEX///$k$///, " is the rationals, reals, complex numbers, or a finite field."},
 	},
     Outputs => {
-        Boolean => {"Whether ", TEX///$\beta$///, " is isotropic"},
+        Boolean => {"whether ", TEX///$\beta$///, " is isotropic"},
 	},
     PARA{"This is the negation of the boolean-valued ", TO2(isAnisotropic,"isAnisotropic"), ". See documentation there."},    
     SeeAlso => {"isAnisotropic", "HilbertSymbol", "signature"}
@@ -18,10 +18,10 @@ document{
     Headline => "Determines whether a Grothendieck-Witt class is anisotropic",
     Usage => "isAnisotropic(beta)",
     Inputs => {
-	GrothendieckWittClass => "beta" => {"Any class ", TEX///$\beta\in\text{GW}(k)$///, " where ", TEX///$k$///, " is the rationals, reals, complex numbers, or a finite field."},
+	GrothendieckWittClass => "beta" => {"denoted by ", TEX///$\beta\in\text{GW}(k)$///, ", where ", TEX///$k$///, " is the rationals, reals, complex numbers, or a finite field"},
 	},
     Outputs => {
-        Boolean => {"Whether ", TEX///$\beta$///, " is anisotropic"},
+        Boolean => {"whether ", TEX///$\beta$///, " is anisotropic"},
 	},
     PARA{"Recall a symmetric bilinear form ", TEX///$\beta$///, " is said to be ", EM "isotropic", " if there exists a nonzero vector ", TEX///$v$///, " for which ", TEX///$\beta(v,v) = 0$///, ". Witt's decomposition theorem implies that a non-degenerate symmetric bilinear form decomposes uniquely into an isotropic and an anisotropic part. Certifying (an)isotropy is then an important computational problem when working with the Grothendieck-Witt ring."},
     PARA{"Over ", TEX///$\mathbb{C}$///, ", any form of rank two or higher contains a copy of the hyperbolic form, and hence is isotropic. Thus we can determine anisotropy simply by a consideration of rank."},