How Good Are You In Algebraic Geometry

2.

If there are two rational functions between two Affine Varieties which are inverse to each other in the region where they are both define. It is said to be

Rational variety
Morphisms
Irrationale
Birationally Equivalent

Correct Answer

A. Birationally Equivalent

Explanation

If there are two rational functions between two Affine Varieties that are inverse to each other in the region where they are both defined, it means that these varieties can be transformed into each other by a series of rational transformations. This implies that the two varieties are birationally equivalent.

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3.

A point of the place belongs to an Algebraic curve if it's coordinates Satisfy a given

Polynomial Equation
Polynomial expression
Regular functions
Affine space

Correct Answer

A. Polynomial Equation

Explanation

The correct answer is "Polynomial Equation." A point of a place belongs to an Algebraic curve if its coordinates satisfy a given polynomial equation. This means that the coordinates of the point can be plugged into the polynomial equation, and the equation will hold true. Algebraic curves are defined by polynomial equations, and the points on the curve are the solutions to these equations. Therefore, a point belonging to an Algebraic curve must satisfy a polynomial equation.

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4.

The fundamental objects of study in Algebraic Geometry are

Algebraic Varieties
Algebraic Functions
Polynomial Equation
Quadratic Equation

Correct Answer

A. Algebraic Varieties

Explanation

Algebraic varieties are the fundamental objects of study in Algebraic Geometry. They are sets of points defined by polynomial equations in several variables. These objects are important in understanding the geometry and properties of solutions to polynomial equations. By studying algebraic varieties, mathematicians can gain insights into the relationships between the solutions of equations and the geometric shapes they represent.

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5.

Expressions containing a number of like terms can be simplified by

Bracketing them
Dividing Through
Multiplying Through
Grouping them together

Correct Answer

A. Grouping them together

Explanation

When we have an expression with a number of like terms, grouping them together means combining the terms that have the same variable or exponent. This simplifies the expression by reducing the number of terms and making it easier to perform operations on them. By grouping like terms together, we can easily combine or cancel out terms with the same variable or exponent, resulting in a simplified expression.

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6.

A natural class of functions on an Algebraic set is called

Rational
Quadrant
Regular functions
Polynomial Equivalent

Correct Answer

A. Regular functions

Explanation

Regular functions are a natural class of functions on an Algebraic set. These functions are defined and continuous on the entire set, and they can be expressed as a quotient of two polynomials. Regular functions are important in algebraic geometry as they provide a way to study the properties of algebraic sets by examining their regular functions.

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7.

An Affine Variety is a rational Variety if it is birationally Equivalent to

Polynomial Equation
Affine space
Quadratic Equation
Equivalency

Correct Answer

A. Affine space

Explanation

An affine variety is considered rational if it is birationally equivalent to an affine space. This means that there exists a rational map between the affine variety and the affine space that is defined by polynomial equations. The concept of rationality in algebraic geometry is closely related to the existence of a birational map between two varieties, which essentially allows for a one-to-one correspondence between points on the varieties. In this case, the correct answer is "Affine space" because it represents the type of variety that an affine variety can be birationally equivalent to in order to be considered rational.

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8.

Just as with the Regular functions on Affine space, the regular functions on V forms a ring called

Functional Ring
Rationality
Birational
Coordinate Ring

Correct Answer

A. Coordinate Ring

Explanation

The regular functions on V form a ring called the coordinate ring. This ring consists of all the functions that are defined on the variety V and can be expressed as a polynomial in the coordinates of V. The coordinate ring is an important concept in algebraic geometry as it allows us to study the geometric properties of V through algebraic techniques.

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9.

As regular maps make the collection of all Affine algebraic set into a category this regular maps are known as

Valencies
Morphisms
Coordinate Rings
Affine maps

Correct Answer

A. Morphisms

Explanation

Regular maps are known as morphisms because they map affine algebraic sets to other affine algebraic sets. In the category of affine algebraic sets, morphisms are the arrows that connect objects (affine algebraic sets) and preserve the structure of the sets. Therefore, regular maps can be considered as morphisms in this category.

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10.

Algebraic Expression can be simplified by Grouping like terms, Removing Brackets and

Expanding Bracket
Enclosing Bracket
Frequency
Dividing Through

Correct Answer

A. Expanding Bracket

Explanation

Expanding the bracket is a method used to simplify algebraic expressions. It involves multiplying each term inside the bracket by the term outside the bracket. This is done by using the distributive property. By expanding the bracket, the expression becomes less complex and easier to work with.

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How Good Are You In Algebraic Geometry

What is important when substituting values into an Algebraic?

Quiz Preview

If there are two rational functions between two Affine Varieties which are inverse to each other in the region where they are both define. It is said to be

A point of the place belongs to an Algebraic curve if it's coordinates Satisfy a given

The fundamental objects of study in Algebraic Geometry are

Expressions containing a number of like terms can be simplified by

A natural class of functions on an Algebraic set is called

An Affine Variety is a rational Variety if it is birationally Equivalent to

Just as with the Regular functions on Affine space, the regular functions on V forms a ring called

As regular maps make the collection of all Affine algebraic set into a category this regular maps are known as

Algebraic Expression can be simplified by Grouping like terms, Removing Brackets and